An evolutionary hybrid cellular automaton model of solid tumour growth

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Journal of Theoretical Biology 246 (2007) 583–603

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An evolutionary hybrid cellular automaton modelof solid tumour growth

P. Gerlee�, A.R.A. Anderson

Division of Mathematics, University of Dundee, Dundee, DD1 4HN, UK

Received 3 June 2006; received in revised form 20 November 2006; accepted 27 January 2007

Available online 12 February 2007

Abstract

We propose a cellular automaton model of solid tumour growth, in which each cell is equipped with a micro-environment response

network. This network is modelled using a feed-forward artificial neural network, that takes environmental variables as an input and

from these determines the cellular behaviour as the output. The response of the network is determined by connection weights and

thresholds in the network, which are subject to mutations when the cells divide. As both available space and nutrients are limited

resources for the tumour, this gives rise to clonal evolution where only the fittest cells survive. Using this approach we have investigated

the impact of the tissue oxygen concentration on the growth and evolutionary dynamics of the tumour. The results show that the oxygen

concentration affects the selection pressure, cell population diversity and morphology of the tumour. A low oxygen concentration in the

tissue gives rise to a tumour with a fingered morphology that contains aggressive phenotypes with a small apoptotic potential, while a

high oxygen concentration in the tissue gives rise to a tumour with a round morphology containing less evolved phenotypes. The tissue

oxygen concentration thus affects the tumour at both the morphological level and on the phenotype level.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Mathematical model; Cellular automaton; Tumourigenesis; Cancer development; Hybrid; Evolutionary dynamics; Clonal evolution; Artificial

neural networks; Micro-environment; Response network

1. Introduction

It is a well-known fact that evolution plays an extensiverole in the development of cancer and that tumours consistof a large number of different subclones that compete forspace and resources (Alexandrova, 2001; Nowell, 1976).Tumour invasion has successfully been modelled by bothcontinuous and discrete mathematical approaches, butmost of these models have failed to capture the evolu-tionary dynamics of tumour growth, and thus neglected avery important aspect of carcinogenesis. The aim of thiswork is to introduce a novel cellular automaton model thatincorporates the evolution of subclones within the tumour.In order to include evolutionary dynamics into the modelthe cells need to be equipped with a genotype, thatdetermines their behaviour, and that is inherited by the

e front matter r 2007 Elsevier Ltd. All rights reserved.

i.2007.01.027

ing author. Tel.: +441382344462; fax: +44 1382 345516.

esses: [email protected] (P. Gerlee),

hs.dundee.ac.uk (A.R.A. Anderson).

daughter cells. This poses problems, as the cell is anextremely complex structure and contains a vast amount ofgenetic material. The solution to this is, as always inmathematical modelling, to simplify the system. Thissimplified model should be straight-forward enough to becomputationally feasible but still complex enough tocapture the interesting dynamics of clonal evolution. Thiswill be done by viewing the genotype of the cell as decisionmechanisms and to focus on a few traits and behavioursthat are often transformed in cancer cells. The decisionmechanism of the cells will be modelled using artificialneural networks (Haykin, 1999) that map the environmentof the cells to a behaviour or action.

1.1. Biological background

Cancer is a genetic disease, which arises from mutationsin single somatic cells. These mutations alter the prolifera-tion control of the cells which leads to uncontrolled celldivision (Hanahan and Weinberg, 2000). The transformed

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ARTICLE IN PRESSP. Gerlee, A.R.A. Anderson / Journal of Theoretical Biology 246 (2007) 583–603584

cells form a neoplastic lesion that may be invasive, in whichcase it is labelled a carcinoma, or benign, an adenoma. Thedecisive factor between these two behaviours is the growthrate and invasiveness of the cells in the neoplasm. Thesetwo properties are in turn driven by what mutations thecells have acquired. In the invasive case the tumour growsin an uncontrolled manner up to a size of approximately106 cells. At this size the diffusion driven nutrient supply ofthe tumour becomes insufficient and the tumour mustinitiate angiogenesis. When the tumour has been vascu-larised the tumour can grow further and at this stagemetastases are often observed. The process of angiogenesishas been modelled successfully by Anderson and Chaplain(1998), but our work will focus on the earlier stages oftumour development.

The purpose of this model is to investigate the pre-vascular stage of invasive tumour growth, when the growthis limited by the diffusion of nutrients. The structure of atumour at this stage is often disordered compared tohealthy tissue. The reason for this is that the cancer cellsare non-responsive to normal growth inhibiting signals(Hanahan and Weinberg, 2000), and thus compete againsteach other for space and nutrients in the tumour. Althoughthe cancer cells have escaped normal growth control, mosttumours exhibit a layered structure, which is due to thediffusion limited supply of nutrients (Sutherland, 1988).When the tumour reaches a critical size the diffusion ofnutrients is not enough to supply the inner parts of thetumour with oxygen, this leads to cell death or necrosis inthe core of the tumour. Outside the necrotic core a rim ofquiescent cells is found and further out a thin rim ofproliferating cells. The mitotic activity therefore only takesplace in a small fraction of the tumour, while the majorityof the tumour consists of cells that are either quiescent ordead. In this competitive environment any cell with agrowth advantage gives that cell and its genotype a higherprobability of surviving. For example cells that have aweaker apoptotic response to hypoxia will have a growthadvantage in the tumour where the oxygen concentration islow. This competition between the cells is similar to theprocess of Darwinian evolution of species (Darwin, 1910)with the exception that the evolution inside a tumour takesplace on a much faster time-scale (Michor et al., 2001). Thereason for this seems to be that most cancer cells aregenetically unstable, which means that they have a muchhigher mutation probability than normal cells (Loeb et al.,2003). One of the main causes of this is that many cancercells have acquired mutations in the p53-gene, which whenunmutated acts as a guardian of the genome and maintainsthe genetic integrity of the cell (Soussi and Lozano, 2005).

Another feature that is often observed in cancer cells istheir tendency to rely on anaerobic metabolism even in theabundance of oxygen, known as the glycolytic phenotype.This is known as the Warburg effect and was discovered inthe 1930s (Warburg, 1930). This behaviour is surprisingbecause the anaerobic metabolic pathway is much lessefficient and the acid that is produced in this pathway is

harmful for the cells, as it lowers the intracellular pH. Butseen from an evolutionary perspective it must lead to someform of growth advantage for the cells that utilise it.Advantages that have been proposed are the possibility tosurvive in poorly oxygenated environments and that theacid produced degrades the surrounding tissue and thusincreases the invasiveness of the tumour (Gatenby andGillies, 2004; Smallbone et al., 2005).The properties that cancer cells acquire at this early stage

of tumour development are mainly driven by the micro-environment in which the cancer cell population evolves.This early selection pressure may also to some extent,determine the later development and invasiveness of thetumour. Recent experimental data has shown that tumourhypoxia is an important factor in tumour developmentbeing directly linked to both tumour morphology (Hockelet al., 1996) and cancer cell aggressiveness (Graeber et al.,1996; Kim et al., 1997). The relationship between tumourmorphology and invasive potential is a known pathologicalfact, however, the evolutionary reasons for this link havestill not fully been understood. The main aim of this paperis therefore to introduce a model capable of capturing thedynamics of clonal evolution and to use this to investigatethe impact of tissue oxygen concentration on both themorphology and the genetic makeup of cancer during theearly avascular stage of development. We therefore hopethis paper will give insight into the effects of tumourhypoxia and its impact on clonal evolution in tumourdevelopment.

1.2. Previous work

Cellular automaton models have been used extensively inmathematical modelling of biological systems (Deutschand Dormann, 2005). The first work using cellularautomata in cancer modelling was done by Duchting andVogelsaenger (1984), who used it to investigate the effectsof radio-therapy. More recent work includes Ferreiraet al.’s (2002) work on tumour morphology, Patel et al.’s(2001) work on the glycolytic phenotype; and Anderson’s(2005) model of tumour invasion. The two former modelsuse a hybrid cellular automaton model, where the CAelements interact with continuous fields of chemicals.Ferreira’s model, which uses a probabilistic updating rulebased on the concentration of nutrients, reproducesvarious tumour morphologies observed in vivo dependingon nutrient consumption. Patel et al.’s work highlightsthe importance of the glycolytic phenotype in cancerinvasion and the impact of vessel density. Both thesemodels use static update rules, which means that all cellsshare the same phenotype. This limits their ability to dealwith phenomena that are not related to evolutionarydynamics. Anderson’s model is somewhat differentfrom a standard CA model as it derives the migratorybehaviour of the cancer cells from a discretisation of aPDE. What more importantly makes this model differentfrom the previous ones is that the cancer cells have

ARTICLE IN PRESSP. Gerlee, A.R.A. Anderson / Journal of Theoretical Biology 246 (2007) 583–603 585

different phenotypes, which allows for the investigation ofevolutionary dynamics.

A three-dimensional cellular automaton approach wasalso used in Kansal et al. (2000a), where the growth ofglioblastoma tumours was investigated. Using a Voronoitesselation with variable grid size they can simulate thetumour growth over several orders of magnitude with goodagreement to experimental data. Further they show thatthe survival probability of a subclone depends on thegrowth advantage it has over other subclones, and that thesurvival probability is non-zero although no growthadvantage exists (Kansal et al., 2000b). In a another studythe above model was extended by introducing a game-theoretic element (Mansury et al., 2006). Assuming that thetumour consists of two distinct subpopulations (oneproliferative and one migratory genotype) the authorsshow that the growth dynamics depends on pay-offs fordifferent cell interactions and that there exists an optimalpay-off for which the tumour velocity (i.e. growth rate) ismaximised. The role of cellular response to the micro-environment was investigated by Mansury et al. (2002) inan agent-based model of brain cancer growth. They showthat the morphology of the resulting tumour depends onhow the cancer cells process information about the micro-environment, and finds a phase transition that separatestwo distinct growth regimes of the tumour, where thecancer cells grow in either small or large clusters.

The evolutionary dynamics of carcinogenesis has alsobeen investigated by Gatenby and Vincent (2003), usingcontinuous techniques from game theory and populationdynamics. Using this approach they identify conditionsnecessary for invasive growth and suggest that the ordinarycytotoxic treatment of the tumour is often unsuccessful dueto the adaptation of the cancer cells to new growthconditions. Another aspect of evolutionary modelling ofcancer is considered in Komarova et al. (2003), a study of astochastic mutation–selection model of early cancer for-mation. Considering mutations in one tumour suppressorgene and chromosomal instability they show that ifchromosomal instability occurs before a mutation, in theconsidered tumour suppressor gene, it becomes a drivingforce in early cancer development.

Artificial neural networks have traditionally been usedfor classification and prediction tasks, some examples are:detection of heart abnormalities (Leung et al., 1990),finger-print recognition (Baxt, 1991) and breast cancerprediction (Floyd et al., 1994). In these tasks the network istrained with a data set that consists of a number ofvariables from each sample together with the outcome ofeach sample (in the case of breast cancer the variables areuniformity of cell size, uniformity of cell shape, marginaladhesion etc. and the outcome is if the breast contains atumour or not). The goal of this procedure is to construct anetwork that will be able to predict the outcome of anunknown sample which was not in the training set. Thereare two approaches to solve this problem, either by using asingle network that is optimised with respect to the training

set using an error minimising algorithm like back-propagation (Haykin, 1999) or by an evolutionary algo-rithm where an evolving population of networks adapts bymutation and selection (Yao, 1993), where the fitness of anetwork is determined by how well it can classify thetraining set. A more recent application of neural networksis to model cell signalling pathways, which was firstsuggested by Bray (1990). He argues that the performanceof cell signalling networks is similar to that of artificialneural networks and that neural networks therefore can beused to model and simulate real signalling pathways.Further he argues that evolution and adaptation ofsignalling pathways occur by small changes in the networkparameters that alter the network connections, and changethe behaviour of the cell. Vohradsky (2001) used a neuralnetwork approach to model the l bacteriophage lysis/lysogeny decision circuit. This model, which incorporatesmultigenic regulation, is in good agreement with experi-mental results, gives further insight into the experimentalobservations and shows that neural networks can success-fully be used to model regulatory pathways.The model presented in this paper is aimed at extending

the present hybrid cellular automaton models and creatinga model that is better suited for the investigation of the roleof evolution in tumour invasion. This will be done bymodelling the regulatory pathways of each cancer cell withan artificial neural network. This approach is of course notthe only way to model clonal evolution, but can be viewedas an alternative to the traditional CA techniquesmentioned above. The increased complexity of this modelis a disadvantage as it renders mathematical analysis of themodel difficult, but this approach also gives some clearadvantages compared to previous CA models. Firstly itallows for a more open-ended model of the evolutionaryprocess where the cells are not confined to a predeterminedset of possible genotypes and secondly it gives a naturalinterpretation of how the micro-environment impacts uponclonal evolution.

2. The model

The basic structure of the model is a two-dimensionalgrid on which the cells reside and represents a slice of thetissue under consideration. The step size of the grid ischosen so that each automaton element is approximatelythe same size as a real cell. Each automaton element caneither be occupied by a cancer cell or be empty. This ofcourse neglects the complex interactions between thecancer cells and the host tissue which may containfibroblasts, macrophages, blood vessels and many othercell types and stimuli. These interactions have been shownto be important factors in tumourigenesis (Rubin, 2003),but in order to keep the model simple and to focus on therole of tissue oxygen concentration on tumour develop-ment we have chosen not to include these aspects in thecurrent model. In order to remove any growth bias thatmight be introduced by explicitly incorporating a vascular

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Hidden Layer

Input

Output

P. Gerlee, A.R.A. Anderson / Journal of Theoretical Biology 246 (2007) 583–603586

oxygen supply to the tumour we have also chosen toexclude any interaction between the tumour and the bloodsupply. Instead the oxygen is supplied from the boundaryof the domain, which corresponds to a situation where aneoplastic lesion develops in a tissue surrounded by bloodvessels. In order to keep track of concentration fields ofoxygen etc., each grid point also holds the concentration ofeach field at that point in space. The concentration fieldsare governed by partial differential equations that describehow the chemicals are produced, consumed, degrade andhow they diffuse.

In previous CA models for cancer growth the automatonrules that govern the evolution of the system have beenstatic, which means that all cancer cells in the modelpossess the same properties. In order to include anevolutionary aspect in our model, each cancer cell ismodelled as an individual entity or agent. This implies thatthe state of an automaton element is not just characterisedby if it holds a cancer cell or not, but by which cell it holds.This means that the update rule for the automaton elementis dependent on the cell that it holds, much like in themodel by Anderson (2005).

2.1. The cell

A cell can be thought of as a computing unit thatgiven a certain input ‘‘calculates’’ an output or response.A classical example of this is when normal epithelial cellsperform apoptosis (programmed cell death) when theyloose adhesion to other cells (Giancotti and Rouslahti,1999). Information from the receptors at the cell surface(i.e. the micro-environment is the input) is transmittedthrough molecular pathways (i.e. determined by thegenotype) and a response is produced (i.e. the phenotypeis the output). Ultimately, the genotype of a cell determineshow it responds to certain stimuli (i.e. the genotype‘‘processes’’ the input and produces an output), and thisresponse can be thought of as the phenotype (see Fig. 1).The abnormal behaviour of cancer cells can be viewed asdisruptions in the response or decision mechanism due togenetic mutations.

In the spirit of this, each cell in the model is equippedwith a decision mechanism, that determines the actions ofthe cell based on the cell genotype, the micro-environmentin which it resides and interactions between these. Thedecision mechanism is subject to mutations during celldivision. The decision mechanism is modelled using an

Micro-

environmentPhenotypeGenotype

Fig. 1. A schematic representation of how a cell takes the micro-

environment as an input which is then ultimately processed by the cell

genotype which in turn decides on an output response which is the

phenotype. The resulting cell phenotype then has the potential to modify

the micro-environment thus setting up a possible feedback loop.

artificial feed-forward neural network (Haykin, 1999),which have been used extensively in solving classificationand recognition problems and have been proven to havethe property of being able to approximate any continuousfunction (Castro et al., 2000). A typical example of a feed-forward neural network can be found in Fig. 2.Including a neural network as a decision mechanism in

each cell of course introduces a higher degree of complexityto the model, but it also makes it possible to model theinteractions between the micro-environment and clonalevolution in a more accurate way. With this approacheach cell will have an individual response based on the localenvironment and how it chooses to process it, compared toconventional cellular automata approaches where thebehaviour of the cells is decided by global parametersthat do not change or evolve e.g. the critical oxygenconcentration below which the cells die. This difference isfundamental to why our approach is suitable for investi-gating clonal evolution, as the underlying mechanism ofclonal evolution and selection is that different sub-clonesbehave in different ways. Another advantage of this modelis that the fitness of a cell is not just defined by its genotype,but is also implicitly defined by the micro-environmentof the cell (cf. Fig. 1). This means that a cell that is adaptedfor hypoxic conditions will have a high ‘‘fitness’’ in apoorly oxygenated environment, but may have a low‘‘fitness’’ in other conditions. The fitness of a cell thusdepends on how it responds and interacts with itsenvironment, a relationship that is difficult to capturewithout including an evolvable response network thatdetermines the behaviour of the cell. Finally the use of aneural network allows for a more open-ended model of theevolutionary process as the cells are not confined to apredetermined set of phenotypes as in Anderson (2005) andmutations can occur that have a range of effects on theresulting phenotype.

Fig. 2. Typical structure of a feed-forward neural network. Information is

given to the input layer and the input is fed through the network via the

hidden layer and produces a response at the output layer.


2.1.1. The network

The response network of the cells (Fig. 3) consists of anumber of nodes that can take real number values. Thenodes are organised into three layers: one input layer, thattakes information from the environment, one hidden layer,and finally an output layer that determines the action of thecell. The nodes in the different layers are connected withvarying weights, determined by two matrices w and W, andthe nodes in the hidden and output layer are equipped withinternal thresholds y and f. The value of the input layer isdetermined by the micro-environment of the cell, thesevalues are then fed through the network and produce aresponse in the output layer that determines the behaviourof the cell (further details of this procedure can be found inthe Appendix).

This neural network approach only serves as an abstractmodel of cellular behaviour, but still shares some featuresof the real signaling and regulatory network of the cell. Theinput layer of the network can be thought of as receptorson the cell surface that interact with extra-cellularmolecules. The connections (weight matrix, w) betweenthe input and hidden layer represents the signaling strengthof these receptors. The hidden layer functions as regulatorygenes that control the behaviour of the cell through theweights of the connection matrix (W) between the hiddenand output layer. Finally the output layer can be thoughtof as the phenotype, as it determines the behaviour of the

Cell Surface

Receptors Genes

Receptor Signalling

Strength

w

ξ1

ξ2

ξ3

ξ4

V1

V2

V3

V4

V5

Oxygen conc.

Glucose conc.

H+ conc.

No. of

neighbours

Fig. 3. The layout of the response network in the cells, the connections betwe

matrix w and the connections between the hidden layer and the output layer a

cell. The connections between the hidden layer and outputlayer may therefore be considered as a mapping from thegenes to the behaviour of the cell, and thus as a mappingfrom the genotype to the phenotype of the cell. With thisanalogy in mind we can think of changing a connectionbetween the input and hidden layer as changing theexpression level of a certain type of receptor and changinga connection between the hidden and output layer asaltering the expression level of a regulatory gene.This network structure is very versatile since there is no

restriction as to what the input nodes represent, they canrepresent any environmental factor we choose to put in themodel. In the most basic setting of our model the onlyinput to the network is the number of neighbours of the celland the local oxygen concentration. The reason for thischoice is that cancer cells often show weaker response tohypoxia-induced apoptosis (Lowe and Lin, 2000) and thatthey tend to adhere less to their neighbours (Cavallaro andChristofori, 2004). In order to investigate the role of theglycolytic phenotype and tumour acidity we will alsoinclude the glucose and hydrogen ion concentration asinputs to the network. This implies that the input vector xwill have four components, x ¼ ðnð~x; tÞ; cð~x; tÞ; gð~x; tÞ;hð~x; tÞÞ, where nð~x; tÞ is the number of neighbours, cð~x; tÞthe oxygen concentration, gð~x; tÞ the glucose concentrationand hð~x; tÞ is the hydrogen ion concentration. The assign-ment of the input nodes can be found in Table 1.

Phenotype

Mapping from Genotype

to Phenotype

WO 1

O 2

O3

O 4

O 5

Proliferation

Quiescence

Apoptosis

Metabolism

Movement

en the input layer and the hidden layer are determined by the connection

re determined by W.

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Table 1

The input and output nodes with corresponding function and variable

name

Node Function Variable

x1 No. of neighbours nð~x; tÞx2 Oxygen concentration cð~x; tÞx3 Glucose concentration gð~x; tÞx4 Acidity [Hþ] hð~x; tÞ

O1 Proliferation P

O2 Quiescence Q

O3 Apoptosis A

O4 Metabolism M

O5 Movement Mov


The output from the network is deterministic and dependsonly on the weight matrices w; W , the threshold vectors y,f and the environmental input vector x. The phenotype ofthe cell is then determined by the values of the nodes in theoutput layer. In our model the output nodes represent theresponse for proliferation, quiescence, apoptosis, move-ment and metabolic pathway. As the first three form agroup of mutually exclusive behaviours (a cell cannotperform these responses simultaneously) the behaviourwith the strongest response is chosen from these three, wecall this the life-cycle response. If the proliferation node hasthe strongest response, the cell divides and produces adaughter cell, if the quiescence node has the strongestresponse the cell remains dormant and if the apoptosisnode is strongest then the cell dies via apoptosis. The cell isallowed to move if the movement response is sufficientlylarge (see Appendix for details). The cell metabolism isdetermined by the response of the metabolic node, and haseither anaerobic or aerobic metabolism depending on thesign of the response (see Appendix for details). Theassignment of the output nodes is summarised in Table 1and the layout of the network can be found in Fig. 3.

2.1.2. Cell metabolism

The metabolism of cancer cells has been shown to differfrom that of normal cells (Dang and Semenza, 1999). Theirmetabolic rate is often higher than normal cells and moreimportantly they have a high tendency to exploit theanaerobic metabolic pathway even in the presence of highoxygen concentrations. This is a somewhat puzzling resultas the anaerobic pathway is much less efficient andtherefore requires a higher uptake of glucose to maintainthe same energy turnover. The two metabolic pathways canbe written in a simplified form as

Aerobic: 1Glucoseþ 6O2! 36ATP,

Anaerobic: 1Glucose! 2ATP.

As can be seen the anaerobic pathway is 18 times lessefficient at creating ATP molecules, which are the energyunits in the cell. Also, both metabolic pathways producehydrogen ions ðHþÞ, but the fact that the anaerobic

pathway produces 18 times less ATP implies that thehydrogen ion production is much higher for the sameenergy turn-over. This high hydrogen ion productionexceeds the metabolic buffering and in order to maintaina normal intracellular pH cells relying on anaerobicmetabolism must transport the excess hydrogen ions tothe extracellular space, which leads to an increased acidityin the surrounding tissue (Robergs et al., 2004).The metabolism of the cancer cells is a crucial aspect of

tumour growth and is therefore important to include in themodel. The switch to anaerobic metabolism is alreadyincluded in the response network of the cell, but we wouldalso like to include variations in glucose and oxygen uptakebetween different subclones in the tumour. For themetabolic pathway the variation in consumption ismodelled by letting the cells utilising the anaerobic path-way consume 18 times more glucose whilst not consumingany oxygen. The difference in metabolism between cellswith different response networks is modelled by taking intoaccount the strength of the network output. The magnitudeof the network output is assumed to be directly propor-tional to the cell metabolism i.e. a high output results in ahigh metabolic rate and vice versa. The logic behind this isthat different amounts of ‘‘energy’’ will be required toperform different network responses e.g. more for pro-liferation and less for quiescence. In order to implementthis we must define a response that corresponds to normalmetabolism and some function that relates the networkresponse to the metabolism. This is done by introducing a‘‘target response’’ Tr and a modulation function F ¼ F ðRÞ,which depends linearly on the response R of the network.The value F of the modulation function is then multipliedby the base consumption/production rates to give themetabolism of each individual cell (see Section 2.2). Theresponse R is set to the highest value among the life-cyclenodes, and thus corresponds to the life-cycle responseperformed by the cell (either P or Q). This is of course acrude way of modelling cell metabolism, but we believe themetabolism to be an important aspect of clonal evolutionand therefore important to include in an evolutionarymodel although in a somewhat simplified manner. Themodulation function is given by

F ¼ maxðkðR� TrÞ þ 1; 0:25Þ, (1)

where k determines the strength of the modulation and theuse of maxð�; 0:25Þ guarantees that the cell has a minimalmetabolism, which is a quarter of the initial one. If thenetwork for example gives an additional response of 0.2,then R ¼ Tr þ 0:2 and the metabolism of that cell will beF ðTr þ 0:2Þ ¼ ð1þ 0:2kÞ times higher than that of a cellwith normal metabolism, i.e. a target response of R ¼ Tr.The parameter k therefore determines to what degree achange in the network response affects the metabolism.Note that R is a function of the input of the network x, butalso of the weight matrices w, W and the thresholds y, f.So R ¼ Rðx;GÞ, where G ¼ ðw;W ; y;fÞ determines thewiring and thresholds of the response network.


The fact that quiescent cells consume less oxygen thanactive cells is introduced by dividing their consumption bya factor q. This factor is not experimentally welldetermined and is assumed to lie somewhere between 2and 100 (Freyer et al., 1984).

2.1.3. Cell death

Cell death can occur in two distinct ways in the model. Ifa cell tries to consume more oxygen or glucose than isavailable at the grid point where the cell resides, it will diefrom starvation (necrosis). The other mechanism is byapoptosis, which occurs if the apoptosis node of thenetwork gets the strongest response.

2.1.4. Cell movement

Cell movement is determined by the movement node inthe response network. As in Anderson (2005) we will usethe notion of an internal adhesion value ai, which is thenumber of neighbours a cell prefers to adhere to. If thenumber of neighbours of a cell is higher than the internaladhesion value ðnð~x; tÞXaiÞ the cell is allowed to move, ifnot it remains stationary. In our network model of cellresponse this means that the input node corresponding tothe number of neighbours will be connected to themovement node, and therefore if nð~x; tÞXai the movementnode takes on a positive value and the cell is allowed tomove.

2.1.5. Proliferation

In order for a cell to divide in the model it has to be inthe proliferative state, this can only happen if the P-nodegets the strongest response among the three life-cycleoutput nodes. The cell also has to have reached theproliferation age in order to divide. If this is the case thecell tries to divide and places a daughter cell at random inone of the four neighbouring automaton elements. If noneof the four CA elements are empty the cell division failsand the cell becomes quiescent. As cancer cells are fastgrowing cells it is a reasonable to assume that most of theenergy metabolised by the cell is used for cell division.Using this assumption we connect the proliferation age tothe metabolism of the cells and model cell proliferation inthe following way: each time a daughter cell is created theproliferation age of that cell is picked randomly fromNðAp;Ap=2Þ normal distribution (taking into account thatthe proliferation age varies among identical cells), where Ap

is the base proliferation age. Every time step the cell isupdated an internal counter X p is incremented with F

(Eq. (1)), and when X p4Ap, the cell has reached itsproliferation age. This means that a cell with a lowproliferation age will have a higher metabolism.

2.1.6. Mutations

The network wiring of the parent cell, which isrepresented by the two matrices w, W and the thresholdsy and f, are copied to the daughter cell under mutations.The number of mutations that occur in the daughter cell

wiring is chosen from a Poisson distribution withparameter p. These mutations are then distributed equallyover the matrices and threshold vectors. The parameter p isthus the average number of mutations per cell division, asthe mean value of a Poisson distribution equals theparameter in the distribution, which in this case is p. Itshould be noted that the mutation rate in this model doesnot correspond to the somatic mutation rate in humancells, as the amount of information copied by a real cell isapproximately 108 orders higher in magnitude. Theincorrect copying is modelled by adding a normaldistributed number s 2 Nð0; sÞ to the daughter cell matrixor threshold entry, which means that x! xþ s, for thoseentries x that are chosen for mutation. The mutations alterthe connection strength between the nodes, which in turnchanges how the cells responds to the micro-environment.If for example a mutation occurs in a connection that linksthe oxygen concentration with the apoptosis node thischanges how the cell responds to the local oxygenconcentration. A detailed description of how the mutationsaffect the behaviour of the cells is given in the Appendix.

2.2. Chemical fields

In the pre-vascular stage of carcinogenesis the tumourhas not yet acquired its own vasculature, the nutrientstherefore have to diffuse from the surrounding bloodvessels to the tumour. The metabolic waste products of thecancer cells also diffuse into the tissue and are transportedaway from the tissue when they reach a blood vessel. Themetabolism of cancer cells includes a large number ofdifferent chemicals that are all needed for maintenance andcell division, but it is known that oxygen and glucoseconcentrations limit the growth of the tumour (Sutherland,1988). We will therefore only incorporate these two fieldsinto the model as well as a field for the hydrogen ionconcentration. A further simplification of the chemicalfields is that we neglect the decay of the nutrients and thatwe include the production of oxygen and glucose in theboundary conditions by applying Dirichlet boundaryconditions with constant functions. These boundaryconditions are meant to imitate a situation were the tissueis surrounded by blood vessels, with constant nutrient andhydrogen ion concentrations, that supply the tumour withnutrients and remove hydrogen ions from the tissue. Thisgives the equations that describe the fields a simple form,similar to those in the models of Patel et al. (2001) andFerreira et al. (2002). The time evolution of the oxygen (2),glucose (3) and hydrogen ion (4) fields are governed by thefollowing set of partial differential equations:

qcð~x; tÞ

qt¼ DcDcð~x; tÞ � f cð~x; tÞ, (2)

qgð~x; tÞ

qt¼ DgDgð~x; tÞ � f gð~x; tÞ, (3)


qhð~x; tÞ

qt¼ DhDhð~x; tÞ þ f hð~x; tÞ, (4)

where Di are the diffusion constants and the f ið~x; tÞ give theindividual cell consumption or production of the chemicali ¼ c; g; h for the cell at position ~x at time t. Theconcentrations of the chemicals are solved on a grid ofthe same step size as the cells, which implies that theconsumption and production terms in (2)–(4) are deter-mined by each individual cell. The f ið~x; tÞ’s are thus definedin the following way,

f ið~x; tÞ ¼

0 if the automaton element at ~x is

empty i.e. no tumour cell at

that lattice point;

riF ð~xÞ if the automaton element is

occupied, i.e. tumour cell

exists at that lattice point;

8>>>>>>>>><>>>>>>>>>:

(5)

where ri are the base consumption/production rates andF ð~xÞ is the modulated energy consumption (1) of theindividual cell occupying the automaton element at ~x.

2.3. Cellular automaton

We have so far described the building blocks thatconstitute the model, and in this section we will describehow all these blocks interact and form the completemodel. The two-dimensional tissue under consideration isrepresented by a N �N grid. The grid is characterised by agrid constant d, which determines the spacing between thecells. The grid points are identified by a coordinate~x ¼ dði; jÞ; i; j ¼ 0; 1; . . . ;N � 1. The chemical concentra-tions interact with the cells according to cellular productionor consumption and are given appropriate initial andboundary conditions. The partial differential equations(2)–(4) are discretised using standard five-point finitecentral difference formulas with space step d and time stepDt. Each time step the chemical concentrations are solvedusing the discretised equations and everyone of the tumourcells are updated in a random order. Every time step of thesimulation each cell is updated according to the schematiclife-cycle flowchart in Fig. 4 and as follows:

(i)
The input vector x is sampled from the localenvironment (i.e. the grid point where the cell resides).
(ii)
A response R ¼ Rðx;GÞ is calculated from the net-work.
(iii)
The cell consumes nutrients according to the actiontaken and the metabolic pathway chosen.
(iv)
The life-cycle action determined by the network iscarried out:� if proliferation (P) is chosen, check if the cell hasreached proliferation age and if there is space for adaughter cell. If both are true the cell divides andthe daughter cell is placed in a neighbouring gridpoint, if not the cell does nothing;
� if quiescence (Q) is chosen the cell becomesquiescent;� if apoptosis (A) is chosen the cell dies;

(v)
If movement is activated and the cells has not divided,it tries to move at random to an empty neighbouringgrid point. If none can be found the cell remainsstationary.
If a cell dies from either apoptosis or necrosis it is no longerupdated. If the cell dies by apoptosis the grid point where itresided is considered empty, but if the cell dies fromnecrosis (starvation) the cell still occupies the grid point.The reason for this is that the two death processes occur indifferent ways. When apoptosis occurs the cell membranecollapses and the cell shrinks, while when necrosis occursthe cell keeps it shape and thus still occupies physical space(Alberts et al., 1994).

2.4. Parameters

The parameters in the model can roughly be divided intothree categories: cell behaviour, tissue structure andmodelling parameters. The first category encompasses allparameters related to the response network and cellularmetabolism, the second relates to the chemical fields andthe third category contains the parameters related to thediscrete nature of the model.One of the most essential groups of cell parameters are

those that define the network wiring and thresholds for theinitial cancer cells, as these cells are used as ‘‘seeds’’ inevery simulation and determine the initial behaviour of thepopulation. As in Anderson (2005) these cells should bedefined in such a way that their behaviour resembles that ofan initial cancer cell phenotype. The response of thenetwork therefore has to capture the essential behaviour ofreal cancer cells. The important features that we want tocapture are:

�
cells should perform apoptosis if the oxygen concentra-tion cð~x; tÞ falls below a certain threshold cap; � cells should die if the glucose concentration gð~x; tÞ falls
below a certain threshold gap;
� cells should not divide if there is no space for the
daughter cell (contact inhibition) i.e. if nð~x; tÞ43;
� cells should perform apoptosis if the acidity hð~x; tÞ is
above a certain threshold hap;
� cells should switch to anaerobic metabolism if the
oxygen concentration cð~x; tÞ falls below cm;
� a cell is allowed to move if the number of neighbours
exceeds the internal adhesion value ai.

The value of cap is difficult to estimate as it depends on thecell type under consideration, but measurement performedin several types of tumours reveal that the oxygenconcentration in the necrotic centre of the tumour is0.5–30% of the concentration in the surrounding tissue(Brown and Wilson, 2004). We therefore estimate cap to be

ARTICLE IN PRESS

Sample the environment

Cell death

Proliferation and

Quiescence

Mitosis

Move to empty space

Calculate Response

Sufficient oxygen?

Prolif-eration?

Quiescence

Reached Proliferati-on Age?

SufficientSpace?

Movement activated?

Free space?

No

Yes

No

No

Yes

Yes

Yes No

Yes

No

Apoptosis

Yes

Necrosis

Start

Fig. 4. Flowchart describing the life-cycle of a cancer cell.

P. Gerlee, A.R.A. Anderson / Journal of Theoretical Biology 246 (2007) 583–603 591

15% of the initial oxygen concentration. The threshold forglucose induced necrosis is set to 50% of the normalglucose concentration, below which hypoglycemia occurs(Ganong, 1999). The acidity threshold is set to match thecritical pH ¼ 7:1 below which cells go into apoptosis(Casciari et al., 1992). The metabolic threshold is set tocm ¼ 0, as we are interested in the emergence of cells thatutilise the anaerobic pathway. Finally we set ai ¼ 3, whichmeans that a cell is allowed to move if it has at least threeneighbours. A phenotype with the above specification was

written by hand and the details of the response network aregiven in the Appendix.If we only look at the number of neighbours and the

oxygen concentration as input to the network the resultingbehaviour of the network can be represented in a two-dimensional plot, which shows the life-cycle response of thenetwork as a function of the two inputs. A plot for theinitial network can be found in Fig. 5, which shows that forlow oxygen ðco0:15Þ concentrations the cell performsapoptosis irrespective of the number of neighbours. For


higher oxygen concentrations ðc40:15Þ the behaviour ofthe cell depends on the number of neighbours, if it exceeds3 the cell becomes quiescent, if not it will proliferate.

2.5. Parameter estimation and non-dimensionalisation

Cancer cells in multi-cell spheroids are known to consumeoxygen at a rate of 2:3� 10�16 mol cells�1 s�1 (Freyer andSutherland, 1986) and we therefore set the base oxygenconsumption rate rc ¼ 2:3� 10�16. The glucose consumptionrates are estimated using the simplified reaction formulas tora

g ¼ rc=6 ¼ 3:8� 10�17 mol cells�1 s�1 for the aerobic path-way and ran

g ¼186� rc ¼ 6:9� 10�16 mol cells�1 s�1 for the

anaerobic, assuming that the amount of energy is indepen-dent of the metabolic pathway. The hydrogen ion productionrate for cells relying on anaerobic metabolism is harder toestimate as it depends on how efficiently the hydrogen ionsare transported through the cell membrane and on tissue

oxygen conc.

no. of neig

hbours

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

Quiescence

Proliferation

A

p

o

p

t

o

s

i

s

Fig. 5. A plot showing the response of the life-cycle nodes as a function of

the oxygen concentration and the number of neighbours. The glucose and

acid concentrations are kept constant at g ¼ 1 and h ¼ 0. At low O2

concentrations co0:15 the cells performs apoptosis (A4Q and A4P) and

for high concentrations c40:15 it proliferates if no3 (P4A and P4Q) or

becomes quiescent if n43 (Q4P and Q4A).

Table 2

A summary of the cell specific parameters in the model

Parameter Meaning Value

rc Base oxygen consumption rate 2.3�10�

rag Aerobic glucose consumption rate 3.8�10�

ranc Anaerobic glucose consumption rate 6.9�10�

rh Hydrogen ion production rate 1.5�10�

Ap Proliferation age 16 h

p Mutation probability 0.01

q Quiescent metabolism factor 5

Tr Target response 0.675

k Modulation strength 6

s Mutation strength 0.25

buffering. We will use rh ¼ 1:5� 10�18 mol cells�1 s�1, thesame as in Patel et al. (2001). The quiescent metabolic factor q

is set to 5, within the experimentally determined range (Freyeret al., 1984). The target response is set to Tr ¼ 0:675. Thisvalue is chosen so that it matches the response of the initialnetwork, giving cells with the initial network consumption/production rates of ri. We have chosen a modulation strengthin (1) of k ¼ 6, because we want to limit the maximummetabolism to be 3 times the base metabolism (similar to thedifference in the least and most aggressive phenotypes inAnderson, 2005) i.e. if the network produces the maximalresponse of R ¼ 1 the metabolism will be approximately 3times higher than the base metabolism. The proliferation age,the time it takes for a cell to move through the entire cell-cycleonce has been experimentally measured to lie between 8 and24h depending the type of cells (Calabresi and Schein, 1993),we use the intermediate value Ap ¼ 16 h. The mutation ratein this model should be related to the somatic mutation ratein human cells, but it should not be compared to the per sitemutation rate, but rather to the number of mutations thatoccur per cell division in genes that are related to the cellularresponses we consider. This is of course very hard to estimateand we therefore set the mutation rate to p ¼ 0:01, whichmeans that on average 1% of the matrix and vector entriesare copied incorrectly. A summary of these cell specificparameters can be found in Table 2 along with appropriatereferences.In order to make the simulation and analysis of the

model simpler we non-dimensionalise Eqs. (2)–(5) inthe standard way. Time is rescaled by the typical time ofthe cell-cycle, t ¼ 16h, and the length by the maximal sizeof a early stage tumour, L ¼ 1 cm. This rescaling of thelength gives each square cell an area of 6:25� 10�6 cm2,which is approximately the same as a cancer cell (Casciariet al., 1992). The chemical concentrations are rescaled usingbackground concentrations: c0 ¼ 1:7� 10�8 molO2 cm

�2

(Anderson, 2005), g0 ¼ 1:3� 10�8 mol cm�2 (Walentaet al., 2001) and h0 ¼ 1:0� 10�13 mol cm�2 (correspondingto normal tissue pH ¼ 7:4) (Patel et al., 2001) and thetumour cell density n0 ¼ 0:0025�2 ¼ 1:6� 105 cells cm�2 (asthe cells reside on a two-dimensional grid). The diffusionconstants for the nutrients and the hydrogen ions have been

Reference

16 mol cells�1 s�1 Freyer and Sutherland (1986)17 mol cells�1 s�1 Calculated from Freyer and Sutherland (1986)16 mol cells�1 s�1 Calculated from Freyer and Sutherland (1986)18 mol cells�1 s�1 Patel et al. (2001)

Calabresi and Schein (1993)

Anderson (2005)

Freyer et al. (1984)

Model specific

Model specific

Model specific

ARTICLE IN PRESS

Table 3

A summary of the micro-environment specific parameters in the model in dimensional units

Parameter Meaning Value Reference

Dc Oxygen diffusion constant 1:8� 10�5 cm2 s�1 Grote et al. (1977)

Dg Glucose diffusion constant 9:1� 10�5 cm2 s�1 Crone and Levitt (1984)

Dc Hydrogen ion diffusion constant 1:1� 10�5 cm2 s�1 Crone and Levitt (1984)

c0 Oxygen background conc. 1:7� 10�8 mol cm�2 Anderson (2005)

g0 Glucose background conc. 1:3� 10�8 mol cm�2 Walenta et al. (2001)

h0 Hydrogen ion background conc. 1:0� 10�13 mol cm�2 Patel et al. (2001)

n0 Cancer cell density 1:6� 105 cells cm�2 Casciari et al. (1992)


measured experimentally and are set to Dc ¼ 1:8�10�5 cm2=s (Grote et al., 1977), Dg ¼ 9:1� 10�5 cm2=s(Crone and Levitt, 1984) and Dh ¼ 1:1� 10�5 cm2=s(Crone and Levitt, 1984). A summary of these micro-environment specific parameters can be found in Table 3along with appropriate references. The new non-dimen-sional variables are thus given by

~~x ¼~x

L; ~t ¼

t

t,

~c ¼c

c0; ~Dc ¼

DctL2

; ~rc ¼tn0rc

c0,

~g ¼g

g0

; ~Dg ¼DgtL2

; ~rg ¼tn0rg

g0

,

~h ¼h

h0; ~Dh ¼

DhtL2

; ~rh ¼tn0rh

h0. (6)

For notational convenience we will drop the tildes on thenon-dimensional variables.

The grid size was set to N ¼ 400, which means that wecan simulate a tumour of radius 200 cells, which if weassume radial symmetry in a three-dimensional settingwould correspond to a tumour consisting of approximately2003 ¼ 8� 106 cells. The time step in the simulation wasset to Dt ¼ 5� 10�4 and the space step to d ¼ 0:0025.

3. Simulations

The model presented here is quite complex and containsmany parameters. Due to this complexity we decided toinvestigate the dynamics of a subsystem of this modelinitially. Specifically we removed cell movement, the choiceof anaerobic metabolism and therefore the chemical fieldsfor glucose and hydrogen ions. The corresponding input andoutput nodes in the cellular response network were alsoremoved. Using this ‘‘minimal’’ system we will investigatehow the oxygen background concentration impacts on thegrowth and evolutionary dynamics of the system.

Note that due to the non-dimensionalisation varying theoxygen background concentration c0 is equivalent to eithervarying the dimensional consumption rate rc or the initialbackground concentration of oxygen in the tissue c0, seeEq. (6). In the following we will refer to changing the

oxygen concentration, but this also corresponds directly tochanging the consumption rate of the cells and thuschanging the cell type of which the tumour consists.In all simulations the initial condition for oxygen is a

homogeneous concentration of cð~x; tÞ ¼ 1 with a constantoxygen concentration of c ¼ 1 on the boundary. Thisboundary condition is meant to imitate a situation wherethe tissue is surrounded by blood vessels that supply thetumour with oxygen via perfusion. Every simulation isstarted with an initial cancer cell population of four cells inthe centre of the grid, each of which has the same initialnetwork (A.4). In the next section we present results thatinvestigate two important aspects of our model: (i) growthdynamics and (ii) evolutionary dynamics.The growth dynamics of the system were investigated for

three different values of the oxygen background concen-tration that represent low to standard oxygen concentra-tion: c0=10, c0=2:5 and c0. We have examined the spatialdistribution of different cell states in the tumour (prolif-erating, quiescent and dead cells) after 33, 66 and 100 timesteps on a grid of size 400� 400. We have also measuredthe time evolution of the total number of cells (includingdead cells) in the tumour and the time evolution of theinvasive distance, which is the distance from the centre ofthe grid (where the tumour starts growing) to the mostdistant cancer cell in the tumour. As the mutationsintroduce randomness into the model both of thesemeasurements were averaged over 20 simulations withdifferent random seeds.The impact of the oxygen concentration on the evolu-

tionary dynamics was investigated by analysing thediversity of the cancer cell populations by looking atthe time evolution of the Shannon index, a measure of thediversity in the population, for oxygen consumption ratesof c0 and c0=10. We also analysed the phenotypes of thetumour cells that emerge in the simulations for the twodifferent values of the oxygen consumption rate.

4. Results

4.1. Growth dynamics

For the lowest oxygen concentration c0=10 the tumourconsists almost exclusively of dead cells and exhibits an

ARTICLE IN PRESS

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Fig. 6. Spatial distribution of the cells at t ¼ 33; 66 and 100 (approx. 22, 44 and 66 days) for c0=10 on a grid of size 400� 400. Proliferating cells are

shown as red, quiescent cells as green, dead cells as blue and empty grid points are white. For this setting hypoxia appears early in the simulation and after

the onset of hypoxia the tumour continues to grow with a fingered morphology due to the low oxygen concentration.

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Fig. 7. Spatial distribution of the cells at t ¼ 33; 66 and 100 (approx. 22, 44 and 66 days) for c0=2:5 on a grid of size 400� 400. Proliferating cells are

shown as red, quiescent cells as green, dead cells as blue and empty grid points are white. For this value of the background oxygen concentration the

hypoxia appears later than for c0=10, but the tumour still exhibits a fingering pattern though less prominent.


obvious fingering morphology (Fig. 6), similar to the sortof structures obtained from diffusion limited aggregation(DLA) (Sander, 1986), growth patterns of bacteria(Matsushita and Fujikawa, 1990) and to the tumourmorphologies in Ferreira et al.’s (2002) model. For thisoxygen concentration cell death appears early in thesimulation at t � 10 (7 days),1 as the diffusion limitedoxygen supply cannot provide the inner parts of thetumour with sufficient oxygen. From this stage the numberof cells in the tumour grows approximately linearly in time(Fig. 9b), with the number of proliferating and quiescentcells remaining constant throughout the simulation. Onlythe cells residing on the boundary of the tumour havesufficient oxygen for proliferation, but even here theoxygen concentration is not enough to sustain normalgrowth as compared with standard oxygen consumption(Fig. 8). Instead the cells have to compete with theirneighbours for oxygen, which ultimately leads to the

1Movies of the simulations are available at hhttp://www.maths.dundee.

ac.uk/�gerlee/CAModel/i.

fingering morphology, where the proliferating cells resideon the tips of the fingers. When a cell divides and places itsdaughter cell outside the existing tumour boundary itreduces the chance for neighbouring cells to survive, as thedaughter cell ‘‘steals’’ oxygen from cells that surround it.This type of growth can be seen as a race towards regionsof higher oxygen concentration.When the oxygen concentration is increased to c0=2:5

the dynamics of the system changes. The growth ofthe tumour is now more regular, although a vaguefingering pattern is still visible (Fig. 7). The lowering ofthe oxygen consumption also changes the temporalbehaviour of the model. The higher oxygen concentrationin the tissue makes it possible for the tumour to growlarger before the centre of the tumour becomes hypoxic.This happens at t � 35 (23 days), after which thetumour continues to grow, but now dominated bydead cells and a thin rim of proliferating cells on theboundary, similar to that seen in Anderson (2005).The asymmetric boundary seen at t ¼ 22 in Fig. 7, andthe absence of a quiescent rim at t ¼ 66 and 100 is due to

http://www.maths.dundee.ac.uk/gerlee/CAModel/



ARTICLE IN PRESS

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Fig. 8. Spatial distribution of the cells at t ¼ 33; 66 and 100 (approx. 22, 44 and 66 days) for c0 on a grid of size 400� 400. Proliferating cells are shown as

red, quiescent cells as green, dead cells as blue and empty grid points are white. The tumour exhibits an almost smooth proliferating boundary with a

homogeneous distribution of dead cells in the interior.

0 10 20 30 40 50 60 70 80 90 1000

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time

inva

sive

dis

tanc

e

0 10 20 30 40 50 60 70 80 90 1000

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6x 10 4

time

tota

l no.

of c

ells

c0/2.5

c0

c0/10

c0/2.5

c0

c0/10

Fig. 9. The time evolution of the (a) invasive distance and (b) total

number of cells, for consumption rates: c0=10, c0=2:5 and c0.


the fact that the cells have lost contact inhibition due tomutations and try to proliferate although there is no spaceavailable.

For the highest oxygen concentration c0 the hypoxia inthe centre of the tumour appears, as expected, at a laterstage compared with the other results. After the onset ofhypoxia the tumour continues to grow in a circular mannerwith a necrotic core and a thin rim of proliferating cells(Fig. 8). The fingering structure found for c0=10 and c0=2:5is now completely gone and the tumour exhibits an almosthomogeneous distribution of dead cancer cells within itsinterior. Again there is an absence of a quiescent rim due tothe loss of contact inhibition.

From Fig. 9 it is clear that the oxygen concentrationaffects the growth rate of the tumour. Both the invasivedistance and the total number of cells grow at the highestrate for the high oxygen concentration, although thedifference is more notable in the time evolution of thetotal number of cells. In order to further investigate howthe oxygen concentration affects the growth dynamics wemeasured how the tumour size depends on the oxygenconcentration in the tissue. This was measured by lettingthe simulation run for t ¼ 100 time steps and counting thenumber of cells present on the grid (including dead cells).This was averaged over 20 runs for each value of theoxygen consumption rate and the result can be found inFig. 10a. In order to measure the impact of mutations onthe growth the above simulations were also performedwithout any mutations ðp ¼ 0Þ and the result can be foundin Fig. 10b.

4.2. Evolutionary dynamics

The Shannon index H (Shannon, 1948) is given by

H ¼ �1

lnðNÞ

XN

i¼0

ðpi ln piÞ, (7)

ARTICLE IN PRESS

0 0.2 0.4 0.6 0.8 1 1.2 1.40

1

2

3

4

5

6

7

8

oxygen concentration

tota

l no. of cells

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

3x 104

x 104

oxygen concentration

tota

l no. of cells

Fig. 10. These plots show the total number of cells with standard

deviation bars after t ¼ 100 time steps (66 days) as a function of the

background oxygen concentration (a) with mutations, (b) without

mutations. In both plots the x-axis is scaled with c0, i.e. c0 ¼ 1.

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

Shannon index

c0 /10

c0

Fig. 11. The time evolution of the Shannon index H for c0 and c0=10. Forc0=10 H increases much faster and reaches a maximum of H � 0:9 at

t � 30 and then slowly decreases, while for c0 it increases at a slower rate

and settles at H � 0:8 after t � 80.


where pi is the probability of finding subclone i in thepopulation and N is the number of distinct subclonespresent in the population. We consider cells to be of thesame subclone if they have identical response networks.The Shannon index reaches its maximum of 1 when allexisting subclones are different (i.e. pi ¼ 1=N for all i), andits minimum 0 when the population consists of only onesubclone. The time evolution of the Shannon index wasmeasured for typical simulations in both settings and theresults can be found in Fig. 11.

In order to measure the behaviour or phenotype of thecells we have devised a measure on the response networkthat quantifies the behaviour of the cell. As already shownin Fig. 5 the response of the network for a two-dimensional

input vector can be visualised by identifying subsets of theinput space with a cellular response. We now measure thefraction of the input space that each of the correspondingresponses occupy. This gives us a three-dimensional vectorS, that we term the response vector, which reflects thebehaviour of the cell. Formally we define three setsxi ¼ fx 2 I ;RðxÞ ¼ ig, where RðxÞ is the network responseto input vector x, i ¼ P;Q;A and I ¼ ½0; 1� � ½0; 4� is theset of all possible inputs to the network. The sizes of thesesubset are now given by

jxij ¼1

B

ZI

di;RðxÞ dx, (8)

where dij is the Kronecker delta (dij ¼ 1 if i ¼ j; 0otherwise) and B ¼ 4 is the area of the entire inputspace. The response vector can now be defined asS ¼ ðjxPj; jxQj; jxAjÞ. The initial cell (Fig. A.1) has ameasure of S ¼ ð0:67; 0:18; 0:15Þ, which means that 67%of the input space corresponds to proliferation, 18% toquiescence and 15% to apoptosis. Note that this measuredoes not give any detailed information about the beha-viour, but rather serves as a measure of the ‘‘average’’behaviour of the cell or the potential the cell has for eachresponse. The measure can also be interpreted as theprobability of a certain response if a random point in theinput space is used as input.For both c0=10 and c0 the system was run for 50 time

steps (30 days) after which a cell from the dominatingsubclone (the most abundant clone at the end of the 50time steps) was extracted from the system. For each cell theresponse vector S and the mean consumption rate (1)(averaged over the entire input space) was calculated. Theresults were averaged over 120 different simulations (withdifferent random seeds) for each of the oxygen concentra-tions and the results can be found in Table 4.

ARTICLE IN PRESS

Table 4

This table shows the average response vector and consumption rate for cells that have evolved under c0 and c0=10

Environment Proliferation Quiescence Apoptosis Consumption

Initial 0.67 0.18 0.15 1.0

c0 0.68 (0.08) 0.17 (0.05) 0.15 (0.03) 1.0 (0.17)

c0=10 0.83 (0.14) 0.11 (0.11) 0.06 (0.06) 1.14 (0.23)

The standard deviations are given in brackets. The response of the initial network is given for comparison.


5. Discussion

From the simulations investigating the growth dynamicsit is obvious that the background oxygen concentrations ofthe tissue affect the growth dynamics of the tumour. In allthe simulations the tumour starts growing consisting ofquiescent cells and a thin rim of proliferating cells. Thisstructure eventually breaks down when cells start dying inthe centre of the tumour because of the low oxygenconcentration. The time at which this happens depends onthe oxygen concentration. As expected this happens earlierfor a low concentration compared to a high concentration.When the oxygen concentration is high the cells can sustaina smoothly growing leading edge that leaves a homo-geneous distribution of dead cells in its wake. In the case oflow concentrations this mechanism breaks down becausethere is not enough oxygen available to support the growthof all the cells on the boundary. This gives rise to acompetition between different subclones in the tumour,where they compete for the limited supply of oxygen. Thelow oxygen concentration also amplifies the advantage ofcells that already have a small growth advantage, as theygrow at the expense of their closest neighbours. If a celldivides too slowly it may get trapped inside the tumour bythe surrounding cells and therefore die out as the oxygenconcentration drops rapidly as the tumour grows. The lowconcentration case is thus more competitive than the highconcentration one, as an inferior phenotype will have ahigher probability of being eliminated in the former case.

The similarity between the observed tumour morpholo-gies and patterns generated by DLA arise because the twoprocesses are in fact quite similar. Both structures are theproduct of a laplacian growth process, where the structuregrows proportional to the gradient of a potential field. Inthe case of DLA the field is the probability distribution ofthe random walkers, and in our model it is the oxygenconcentration field, which in the limit of low oxygenconcentration approximates a potential field.

As seen in Fig. 9 the oxygen concentration influences thespeed at which the tumour grows. The oxygen concentra-tion of the tissue affects both the time evolution of theinvasive distance (Fig. 9a) and total number of cells(Fig. 9b). A higher oxygen concentration gives rise to atumour that grows faster with respect to both invasivedistance and total number of cells. The reason for this isthat a low oxygen concentration reduces the amount of

oxygen available at the tumour boundary. This naturallyleads to a reduced growth of the tumour as the cells rely onoxygen for survival and proliferation. This behaviour isquite expected, but what is interesting to observe is that theinvasive distance does not depend as strongly on theoxygen concentration as the total number of cells. Theaverage total number of cells after 100 time steps isapproximately 5� 104 cells for c0 and 1� 104 cells forc0=10, giving a ratio of approximately 5. As the cells growin a single layer the total number of cells is proportional tothe area of the tumour, which for a compact tumour isproportional to R2, the radius squared. If the tumoursfrom c0 and c0=10 shared the same compact morphologythen the ratio between the invasive distance of c0 and c0=10would be

ffiffiffi5p� 2:24, as the invasive distance is approxi-

mately the radius of the tumour. But the ratio is in fact160110� 1:45 and the reason for this is that the fingered

morphology does not grow as a compact structure, butrather as a ‘‘fractal’’ object (Fig. 6) where the total numberof cells scales as the Ra, where 1oao2 (Mandelbrot, 1977).In fact we calculated a � 1:6 for c0=10, which means thatalthough the growth rate of the total number of cells ismuch lower than for c0, the fractal morphology gives thetumour a considerable invasive distance.If we now turn to Fig. 10 we see as expected that the

tumour size is smaller as the oxygen concentrationdecreases, but what is interesting to observe is how themutations affect the growth dynamics. The simulationsperformed with mutations give rise to tumours that areapproximately twice the size compared to simulations withcorresponding concentrations but without mutations. Thereason for this is that the cells acquire mutations thatincrease their capability to proliferate and which reducetheir proliferation age, both which clearly increase thegrowth rate of the tumour. What we also can observe isthat the standard deviations depend on the oxygenconcentration when mutations are present (Fig. 10a). Thevariability of the simulation results are significantly higherfor high oxygen concentrations compared to the low, butobviously when no mutations (Fig. 10b) occur thisbehaviour disappears. This effect occurs because theselection pressure is different in the low and highconcentration cases. A tumour growing in a tissue with ahigh oxygen concentration can grow at different ratesbecause the environment in which it grows is not veryharsh. The selection pressure on the population is therefore


weak, which implies that the tumour may consist of cellswith a low growth rate, which in turn leads to a smallertumour. In the case of a tumour growing in a hypoxicenvironment the selection pressure is stronger as the micro-environment is now harsher. This means that only cellswith a high growth rate will survive, and thus the tumourwill always consist of fast growing cells and exhibit a lowervariance in the size.

The oxygen concentration also influences the timeevolution of the Shannon index. Fig. 11 displays twoqualitatively different behaviours in the time evolution ofthe index. In the low concentration setting H increases fastand reaches a maximal value of H � 0:9 after about 30time steps (20 days). It then decreases slowly until it reachesa value of H � 0:8 at the end of the simulation (65 days). Inthe high concentration case H increases at a lower pace,but settles around the same value of H � 0:8 at the end ofthe simulation. The explanation for the different beha-viours can be found in the morphology of the tumours. Inthe low concentration case the living cells reside on the tipsof the fingered tumour. As there is no contact betweenthese tips they are essentially isolated colonies of cells thatevolve independently. This structure facilitates a higherdiversity in the population and as a result we see a fastincrease in the population diversity. In the high concentra-tion case, which has a more regular growth, the morphol-ogy of the tumour allows for a higher degree ofcompetition between different subclones. This competitionleads to lower diversity, as some subclones are eliminatedby superior ones. As a result the Shannon index increasesslower than in the high consumption case. These observa-tions are in agreement with experimental studies ofdiversity in bacterial colonies that grow in structured andunstructured environments (Korona et al., 1994). What isinteresting to observe is that the increase in the Shannonindex correlates with the appearance of cell death in thetumour. In the low concentration case cell death appearsalmost instantaneously in the simulation and we conse-quently see an early increase in the Shannon index. In thehigh concentration case the tumour consist of a core ofquiescent cells with a rim of proliferating cells for to40.During this time the Shannon index remains almostconstant at H � 0:2 (after an initial increase). When celldeath appears at t � 40 the Shannon index starts toincrease and eventually settles at approximately the samevalue as for c0=10. The appearance of cell death changesthe selection pressure on the population, and makes itpossible for subclones to exploit the new environmentand we consequently observe an increase in the popula-tion diversity. For both settings H settles approximatelyaround the same value H � 0:8. This value dependson evolutionary settings such as the mutation probabilityand mutation strength, which are identical for the twocompared here.

The two settings also give rise to different phenotypes.The averages shown in Table 4 display a significantdifference between the cells that have evolved in the high

and low oxygen concentrations. The dominant subclonesfrom the high concentration differ little from the initialcells, the only difference is a slight increase in proliferationpotential and decrease in quiescence potential. Theapoptosis potential is the same as for the initial cells andthe consumption rate is also unchanged. For the cells fromthe low concentration on the other hand we can see anobvious difference. They have on average increasedproliferation potential and consequently decreased quies-cence and apoptosis potentials. We also see an increase inthe consumption rate, which also means a decrease in theproliferation age. The cells from the high consumptionsettings have thus evolved further away from the initial cell.They can also be said to be more aggressive as they will beproliferative in a larger part of the input space and thus in awider range of environmental conditions.The reason for this is that a lower oxygen concentration

in the tissue creates an environment which is harsher andtherefore more competitive, and as a result the selectionpressure on the population is stronger. As mentionedearlier the growth in the low concentration case can beviewed as a race towards regions of higher oxygenconcentration and it is this mechanism that makes theselection in the high consumption case so strong. If a cellcannot go into mitosis because the oxygen concentration istoo low or because it has not yet reached it’s proliferationage it might be trapped inside the tumour, and as theoxygen concentration drops rapidly over the tumourboundary the cell will eventually die. It is therefore aselective advantage to have a large proliferation potentialand to have a low proliferation age, exactly what weobserve in the cells from the high consumption simulations.The high concentration case also exhibits competitionbetween the cells, but as cell death appears later in thesimulation and the oxygen concentration does not decay asrapidly over the tumour boundary, the selective forces areless evident in this case. What is also interesting to observeis that the cells from the low concentration sacrifice normalconsumption for a low proliferating age, as the two arelinked through the response modulation (1). This meansthat the proliferation age is under stronger selectivepressure than the consumption rate. Although a highconsumption increases the possibility of death by starva-tion this is over-shadowed by the advantage of a faster cellcycle.Combining the results from the growth and evolutionary

dynamics we can observe that there is a clear connectionbetween the morphology of the tumour and the phenotypesit contains. The high concentration case gives rise to around tumour morphology and phenotypes with a lowproliferation potential, while the low concentration casegives rise to a fingered morphology containing aggressivecells with a higher proliferation potential and faster cellcycle. This implies that the oxygen concentration (or baseconsumption rate) affects the tumour on both a globalmorphological level and on the phenotypic level of theindividual cells.


If we compare the results from this model with theresults from Anderson (2005) and Anderson et al. (2006)we see that although the two models are quite different,both with respect to how cell behaviour and mutations aremodelled, they give similar results. Anderson’s model usesa simplistic model for mutations, where the mutations areeither linear (the phenotypes become progressively moreaggressive) or completely random, whereas in this modelthe mutations are random but the daughter cells are alwayssimilar to their parents as they inherit the responsenetwork. In Anderson’s model the tumour growth isdriven by a combination of proliferation and migrationvia haptotaxis in response to gradients within the extra-cellular matrix (ECM), while in our model growth onlyoccurs by proliferation. Despite these differences bothmodels suggest that an aggressive phenotype is more likelyto appear in a tumour exhibiting a fingered morphology. InAnderson’s model the morphology is heavily influenced bythe structure of the ECM, where a heterogeneous ECMgives rise to a fingered tumour containing aggressivephenotypes. This heterogeneous ECM could be interpretedas a harsh environment. In this model it is the tumourhypoxia that drives the fingering morphology, but weobserve the same connection between morphology andcancer cells aggressiveness as in Anderson’s model.

Overall the model results make some interesting predic-tions about links between tumour aggressiveness, morphol-ogy, genetic heterogeneity and the micro-environment.Some of these predictions are in agreement with recentexperimental results, in particular Graeber et al. (1996),Kim et al. (1997) show that cancer cells from hypoxictumours have a smaller apoptotic potential. The hypoxicmicro-environment provides a selection pressure in thetumour for subclones which can withstand the low oxygenconcentration. This is precisely what we observe in ourmodel i.e. a low oxygen concentration (a higher degree ofhypoxia) gives rise to phenotypes with a smaller apoptoticpotential. In another experimental study on lung carcino-ma cells, Koshikawa et al. (2006) show that sub-clones withsmaller apoptotic potential have a growth advantage inhypoxic tumours, which also supports our findings.Tumour hypoxia has been linked to tumour morphologyin clinical studies (Hockel et al., 1996) and more recently inan experimental study (Pennacchietti et al., 2003). In theclinical study on cervical cancers, it was shown thattumours with a high degree of hypoxia exhibited largertumour extensions compared to well-oxygenated tumoursof similar clinical stage and size. Additionally, they showthat patients with these same hypoxic tumours had asignificantly worse survival probability compared topatients with non-hypoxic tumours. In the experimentalstudy it was found that tumour spheroids grown incollagen gels under hypoxic conditions produced fingeredinvasive tumour morphologies, in contrast to the sphericalstructures that grew under normoxic conditions. Both ofthese in vivo and in vitro results show a clear correlationwith our in silico simulations, in particular we show that a

low oxygen concentration (a higher degree of hypoxia)gives rise to a fingered morphology and that these fingeredtumours contain a more aggressive population (thus morelife threatening).Whilst the experimental evidence discussed above is in

good agreement with the predictions from our model, it isclear that a more explicit experimental approach would berequired to fully validate the model predictions. One suchexperiment would be to seed tumours in vivo in differentmouse tissues that are known to have distinct levels ofoxygenation. The morphology of the resulting tumourswhich grow could then be measured. By additionally takingbiopsies from the tumour, experimental assays could thenbe used to measure both the proliferative and apoptoticpotential of the tumour. This type of experimentalprocedure would be able to give results on both themacroscopic level of morphology and on the phenotypelevel of individual cells simultaneously, which could theneasily be compared with the results from our model.

6. Conclusions

In this paper we have presented a novel hybrid cellularautomaton model which is aimed at investigating theevolutionary dynamics of early tumour development.Simulations performed with a subsystem of this modelshow that the tissue background oxygen concentration (orbase oxygen consumption rate) affects both the growthdynamics of the tumour and the evolutionary dynamics ofthe competition between different subclones within thetumour.The results reveal that there is a connection between the

morphology of the tumour and the phenotypes that itcontains. The fingered morphology, which occurs when thetumour grows in a tissue with a low background oxygenconcentration ðc0=10Þ, contains phenotypes that haveevolved further away from the ancestral cell and thatexhibit a more aggressive phenotype compared to tumourgrown under normoxic conditions ðc0Þ, that grows with around morphology. The oxygen concentration also influ-ences the population diversity of the cancer cells. Weobserve a higher heterogeneity in the population in harshgrowth conditions, where the oxygen concentration is low,compared to the high concentration case. Although this isthe case it should be noted that a harsh environment givesrise to a lower variance in the tumour size, which suggeststhat the evolution is more directed towards aggressivephenotypes in the low concentration case.Tumour morphology under hypoxic conditions has

already been investigated by Ferreira et al. (2002) andseveral previous models have investigated clonal selection(Kansal et al., 2000b; Mansury et al., 2006; Gatenby andVincent, 2003), but crucially it is the link betweenmorphology and clonal evolution that is the central resultof this paper. The model presented here is quite complex,but it is this complexity that makes it possible to obtainthe results presented in the paper. As mentioned in the


introduction this approach has both advantages anddisadvantages, but based on our results we believe thatthe former out-weighs the latter and that our model cangive important insights into the dynamics of tumourgrowth and evolution.

The results from our model show that the micro-environment in which the tumour grows is very much thedriving force of the type of tumour that will evolve bothgenetically and morphologically (Anderson, 2005). Thisimplies that a tumour which grows in a tissue that is poorlyoxygenated or consisting of cells with a high oxygenconsumption rate is more likely to exhibit a fingeringmorphology and to contain aggressive cancer cells. Ofcourse the avascular growth regime only spans a short timein the growth of a tumour, but the results from our modelhighlight the importance of the conditions under which thisinitial growth occurs. This could have implications forcancer treatment, where it might be possible to prevent theevolution of aggressive cancer cells by altering the micro-environment in which the tumour grows. Intriguingly italso implies that the use of anti-angiogenic treatment mightactually increase the invasiveness of the tumour by creatinga harsh micro-environment that destabilises the tumourmorphology (Cristini et al., 2005) and creates a selectionpressure that favours aggressive phenotypes. Insteadtreatment should aim at normalising the tissue micro-environment which would lead to a non-invasive tumourwith a well-defined boundary that is less likely to containaggressive phenotypes.

Acknowledgements

This work was funded by the National Cancer Institute,Grant Number: U54 CA 113007-02.

Appendix A

Here we discuss in detail the structure of the responsenetwork, describe the network parametrisation and explainprecisely how mutations change the behaviour of the cells.

A.1. Mathematical formulation of the response network

The network consists of a number of nodes that can takereal number values. The nodes are organised into threelayers: the input layer x, the hidden layer V , which can beinterpreted (for our model) as the processing layer, and theoutput layer O. The nodes in the different layers areconnected via links, of varying connection strengths, whichcan modify the value of each node via the transfer function(TðxÞ, below). The connections between the input layer andthe hidden layer are defined by a connection matrix w,where wij determines the connection strength between nodej in the input layer ðxjÞ and node i in the hidden layer ðViÞ.Likewise the connections between the hidden layer and theoutput layer are defined by a connection matrix W, whereW ij determines the connection strength between node j in

the hidden layer ðV jÞ and node i in the output layer ðOiÞ.The nodes in the hidden and output layer are also equippedwith thresholds, where yi is the threshold of node i in thehidden layer ðV iÞ and fj is the threshold of node j in theoutput layer Oj. These are real valued parameters used tomodulate the impact of the connections to each node.The response of the network for a given input vector is

calculated in the following way: first the values of the inputnodes are set to the input vector x ¼ ðnð~x; tÞ; cð~x; tÞ; gð~x; tÞ;hð~x; tÞÞ. The node values of the input layer are then fed tothe hidden layer using a standard transfer function, TðxÞ,and the connection matrix w.

TðxÞ ¼1

1þ e�2x. (A.1)

This is a standard function used in neural networks(Haykin, 1999) and guarantees that the resulting nodevalues of the hidden layer are in the range ½0; 1�. The valueof node j in the hidden layer is given by

Vj ¼ TX

k

wjkxk � yj

!, (A.2)

which is the sum of the input nodes weighted with theconnection matrix w and the threshold yj. The sameprocedure is repeated for the output layer, which meansthat value of node i in the output layer is given by

Oi ¼ TX

j

W ijVj � fi

!

¼ TX

j

W ijTX

k

wjkxk � yj

!� fi

!. ðA:3Þ

The behaviour of the cells is then determined by thevalues of the nodes in the output layer. From (A.3) wecan see that the value of each output node is a functionof all input nodes and that the impact of each inputnode depends on the network parameters w; W ; y and f.Therefore, the behaviour of the network is determinedby these parameters. When a cell divides, to modelmutation, we allow these parameters to be copied tothe daughter cells with a small variation (see below forfurther detail).

A.2. Movement and metabolism

Unlike the life-cycle response nodes which are linked(proliferation, quiescence and apoptosis) the movementand metabolism responses are independent and can occursimultaneously. A cell is allowed to move if the movementresponse is above 0.5, which corresponds to a positiveinput to the movement node (as Tð0Þ ¼ 0:5). The switchbetween aerobic and anaerobic metabolism is determinedby the response of the metabolic node, if the input to themetabolism node is negative the cell performs anaerobicmetabolism and aerobic if the input is positive.


A.3. Network parametrisation

The response network of the initial cell population inshape of the connection matrices and threshold vector is asfollows:

w ¼

1 0 0 0

0:5 0 0 0

0 �2 0 0

0 0 �2 0:5

1 0 0 0

0BBBBBBBB@

1CCCCCCCCA,

W ¼

�0:5 1 �0:5 0 0

0 0:55 �0:5 0 0

0 0 2 2 0

0 0 0 0 0

0 0 0 0 1

0BBBBBBBB@

1CCCCCCCCA,

y ¼ ð0:55 0 0:7 � 0:25 0Þ,

f ¼ ð0 0 0 0 0:75Þ. ðA:4Þ

These matrices are quite sparse and gives rise to a responsenetwork that can be seen in Fig. A.1. This network satisfiesthe basic specifications given above (see Section 2.4) for areal cancer cell, but is in no way the only network that doesso. A network with different wiring and thresholds couldgive the same response, but what is important to stress hereis that we are interested in studying the evolution of the

0.55

0

0.7

-0.25

0

No. of neighbours

Oxygen conc.

Glucose conc.

H+ conc.

1

0.5

-2

1

0.5

-2

Fig. A.1. The wiring of the initial cell’s network. The weights of the connection

the nodes.

network, which is largely independent of the initialstructure. The network weights and thresholds are there-fore chosen only to produce the correct response, not tomimic any specific signalling pathways. Another approachcould be to evolve the initial network from a population ofrandom networks using the desired response as the trainingset (Yao, 1993).

A.4. Mutations

Mutations in the model occur during cell division whenthe network parameters of the parent cell are copied to thedaughter cell and are modelled by adding a normalrandomly distributed number to the parameters chosenfor mutation. In order to understand how differentmutations affect the phenotype of the cell we now discussthe impact of mutations upon the initial network (A.4).First consider a mutation in w11: an increase in thisconnection will reduce the response (A.1) of the prolifera-tion node as a function of the number of neighbours (as thesecondary connection W 11 is negative), see Fig. A.1. On theother hand if w11 is decreased the proliferation response(A.1) will be stronger, as x1 ¼ nð~x; tÞ (the no. of neigh-bours) this corresponds to a cell with a weaker response tocontact inhibition.Now consider a mutation in W 33, which connects oxygen

concentration and apoptosis. An increase in this weightwill reduce the response (A.1) to apoptosis as a function ofthe oxygen concentration, while a decrease in this weightwill increase the dependence of oxygen on apoptosis. A cell

0

0

0

0

0.75

-0.5

1

-0.50.55

2

1

Proliferation

Quiescence

Apoptosis

Metabolism

Movement

2

-0.5

s are given next to the arrows and the node thresholds are displayed inside


with a lower dependence of oxygen on apoptosis willclearly have an advantage in hypoxic regions and willconsequently have more of an opportunity to proliferate.This highlights the fact that the fitness of the cell dependson both the phenotype of the cell and the environment inwhich lives (i.e. the fitness of the cell is implicit).

From these two examples we can see that mutations indifferent parts of the network will affect different aspects ofthe cellular behaviour. Additionally, it is important torealise that mutations can also occur at links that areinitially set to zero. For example a sub-clone that acquires alink between the oxygen concentration and the choice ofmetabolic pathway may emerge. Such a sub-clone wouldhave a growth advantage in poorly oxygenated regions,even though this was not predefined in the model. Thishighlights the fact that we model clonal evolution in anopen-ended manner, which is an important aspect of thetrue evolutionary process.

References

Alberts, B., Bray, D., Lewis, J., Raff, M., Roberts, K., Watson, J., 1994.

Differentiated cells and the maintenance of tissue. In: The Cell, third

ed. Garland Publishing, New York, pp. 1173–1175 (Chapter 22).

Alexandrova, R., 2001. Tumour heterogeneity. Exp. Pathol. Parasitol. 4,

57–67.

Anderson, A.R.A., 2005. A hybrid mathematical model of solid tumour

invasion: the importance of cell adhesion. Math. Med. Biol. 22,

163–186.

Anderson, A.R.A., Chaplain, M., 1998. Continuous and discrete

mathematical models of tumor-induced angiogenesis. Bull. Math.

Biol. 6, 857–899.

Anderson, A.R.A., Weaver, A.M., Cummings, T.M., Quaranta, V.,

2006. Tumor morphology and phenotypic evolution driven by

selective pressure from the microenvironment. Cell, doi:10.1016/

j.cell.2006.09.042.

Baxt, W., 1991. Use of an artificial neural network for the diagnosis of

myocardial infarction. Ann. Intern. Med. 115, 843–848.

Bray, D., 1990. Intracellular signalling as a parallel distributed process.

J. Theor. Biol. 143, 215–231.

Brown, J., Wilson, W., 2004. Exploiting tumour hypoxia in cancer

treatment. Nature 4, 437–447.

Calabresi, P., Schein, P.E., 1993. Medical Oncology, second ed. McGraw-

Hill, New York.

Casciari, J.J., Sotirchos, S.V., Sutherland, R.M., 1992. Variation in

tumour cell growth rates and metabolism with oxygen-concentration,

glucose-concentration and extra-cellular pH. J. Cell. Physiol. 151,

386–394.

Castro, J.L., Mantas, C.J., Benitez, J.M., 2000. Neural networks with a

continuous squashing function in the output are universal approx-

imators. Neural Networks 13, 561–563.

Cavallaro, U., Christofori, G., 2004. Cell adhesion and signaling by

cadherins and Ig-CAMS in cancer. Nat. Cancer Rev. 4, 118–132.

Cristini, V., Frieboes, H., Gatenby, R., Caserta, S., Ferrari, M., Sinek, J.,

2005. Morphologic instability and cancer invasion. Clin. Cancer Res.

11, 6772–6779.

Crone, C., Levitt, D.G., 1984. The cardiovascular system. In: Handbook

of Physiology: A Critical, Comprehensive Presentation of Physiologi-

cal Knowledge and Concepts. American Physiological Society,

Bethesda, MD, pp. 414, 434–437 (Chapter 2).

Dang, C., Semenza, G., 1999. Oncogenic alterations of metabolism.

Trends Biochem. Sci. 24, 68–72.

Darwin, C., 1910. On the Origin of Species. Ward, Lock and Co.

Deutsch, A., Dormann, S., 2005. Cellular Automaton Modeling of

Biological Pattern Formation. Birkhauser, Boston.

Duchting, W., Vogelsaenger, T., 1984. Analysis, forecasting and control of

three-dimensional tumor growth and treatment. J. Med. Syst. 8,

461–475.

Ferreira, S.C., Martins, M.L., Vilela, M.J., 2002. Reaction–diffusion

model for the growth of avascular tumor. Phys. Rev. E 65, 021907.

Floyd, C., Lo, J., Yun, A., Sullivan, D., Kornguth, P., 1994. Prediction of

breast cancer malignancy using an artificial neural network. Cancer 74,

2944–2998.

Freyer, J.P., Sutherland, R.M., 1986. Regulation of growth saturation and

development of necrosis in EMT6/Ro multicellular spheroids by the

glucose and oxygen supply. Cancer Res. 46, 3513–3520.

Freyer, J.P., Tustanoff, E., Franko, A.J., Sutherland, R.M., 1984. In situ

consumption rates of cells in v-79 multicellular spheroids during

growth. J. Cell. Physiol. 118, 53–61.

Ganong, W., 1999. Review of Medical Physiology, 19th ed. Appleton &

Lange, New York, p. 329 (Chapter 19).

Gatenby, R., Vincent, T., 2003. Application of quantitative models from

population biology and evolutionary game theory to tumor therapeu-

tic strategies. Mol. Cancer Ther. 2, 919–927.

Gatenby, R.A., Gillies, R.J., 2004. Why do cancers have high aerobic

glycolysis? Nat. Rev. Cancer 4, 891–899.

Giancotti, F., Rouslahti, E., 1999. Integrin signalling. Science 285,

1028–1032.

Graeber, T., Osmanian, C., Jacks, T., Housman, D., C.J., K., S.W., L.,

Giaccia, A., 1996. Hypoxia-mediated selection of cells with diminished

apoptotic potential in solid tumours. Nature 379, 88–91.

Grote, J., Susskind, R., Vaupel, P., 1977. Oxygen diffusivity in tumor

tissue (DS-carcinosarcoma) under temperature conditions within the

range of 20–40 C. Pflugers Arch. 372, 37–42.

Hanahan, D., Weinberg, R., 2000. The hallmarks of cancer. Cell 100,

57–70.

Haykin, S., 1999. Neural Networks: A Comprehensive Foundation,

second ed. Prentice-Hall, Englewood Cliffs, NJ.

Hockel, M., Schlenger, K., Aral, B., Mitze, M., Schaffer, U., Vaupel, P.,

1996. Association between tumor hypoxia and malignant

progression in advanced cancer of the uterine cervix. Cancer Res. 56,

4509–4515.

Kansal, A.R., Torquato, S., Harsh IV, G.R., Chiocca, E.A., Deisboeck,

T.S., 2000a. Simulated brain tumor dynamics using a three-dimen-

sional cellular automaton. J. Theor. Biol. 203, 367–382.

Kansal, A.R., Torquato, S., Chiocca, E.A., Deisboeck, T.S., 2000b.

Emergence of a subpopulation in a computational model of tumor

growth. J. Theor. Biol. 207, 431–441.

Kim, C., Tsai, M., Osmanian, C., Graeber, T., Lee, J., Giffard, R.,

DiPaolo, J., Peehl, D., Giaccia, A., 1997. Selection of human cervical

epithelial cells that possess reduced apoptotic potential to low-oxygen

conditions. Cancer Res. 57, 4200–4204.

Komarova, N., Sengupta, A., Nowak, M., 2003. Mutation–selection

networks of cancer initiation: tumor suppressor genes and chromoso-

mal instability. J. Theor. Biol. 223, 433–450.

Korona, R., Nakatsu, C., Forney, L., Lenski, R., 1994. Evidence of

multiple adaptive peaks from populations of bacteria evolving in a

structured habitat. Proc. Natl Acad. Sci. 91, 9037–9041.

Koshikawa, N., Maejima, C., Miyazaki, K., Nakagawara, A., Takenaga,

K., 2006. Hypoxia selects for high-metastatic lewis lung carcinoma

cells overexpressing MCL-1 and exhibiting reduced apoptotic potential

in solid tumors. Oncogene 25, 917–928.

Leung, M., Engeler, W., Frank, P., 1990. Fingerprint processing using

back-propagation neural networks. In: Proceedings of the Interna-

tional Joint Conference on Neural Networks I, vol. 1, pp. 15–20.

Loeb, A., Loeb, K.R., Anderson, J.P., 2003. Multiple mutations and

cancer. Proc. Natl Acad. Sci. 100, 776–781.

Lowe, S.W., Lin, A.W., 2000. Apoptosis in cancer. Carcinogenesis 21,

485–495.

Mandelbrot, B., 1977. Fractals: Form, Chance and Dimension. Freeman,

San Francisco.

http://10.1016/j.cell.2006.09.042

http://10.1016/j.cell.2006.09.042


Mansury, Y., Diggory, M., Deisboeck, T., 2006. Evolutionary game

theory in an agent-based brain tumor model: exploring the ‘genoty-

pe–phenotype’ link. J. Theor. Biol. 238, 146–156.

Mansury, Y., Kimura, M., Lobo, J., Deisboeck, T., 2002. Emerging

patterns in tumor systems: simulating the dynamics of multicellular

clusters with an agent-based spatial agglomeration model. J. Theor.

Biol. 219, 343–370.

Matsushita, M., Fujikawa, H., 1990. Diffusion-limited growth in bacterial

colony formation. Physica A 168, 498–506.

Michor, F., Iwasa, Y., Nowak, M.A., 2001. Dynamics of cancer

progression. Nat. Rev. Cancer 4, 197–205.

Nowell, P.C., 1976. The clonal evolution of tumour cell populations.

Science 194, 23–28.

Patel, A., Gawlinski, E.T., Lemieux, S.K., Gatenby, R.A., 2001. A cellular

automaton model of early tumor growth and invasion: the effects of

native tissue vascularity and increased anaerobic tumor metabolism.

J. Theor. Biol. 213, 315–331.

Pennacchietti, S., Michieli, P., Galluzzo, M., Mazzone, M., Giordano, S.,

Comoglio, P.M., 2003. Hypoxia promotes invasive growth by transcrip-

tional activation of the met protooncogene. Cancer Cell 3, 347–361.

Robergs, R., Ghiasvand, F., Parker, D., 2004. Biochemistry of exercise-

induced metabolic acidosis. Am. J. Physiol. Regul. Integr. Comp.

Physiol. 287, 502–516.

Rubin, H., 2003. Microenvironmental regulation of the initiated cell. Adv.

Cancer Res. 90, 1–62.

Sander, L.M., 1986. Fractal growth processes. Nature 322, 789–793.

Shannon, C.E., 1948. A mathematical theory of information. Bell Syst.

Tech. J. 27, 379–423.

Smallbone, K., Gavaghan, D.J., Gatenby, R.A., Maini, P.K., 2005. The

role of acidity in solid tumour growth and invasion. J. Theor. Biol.

235, 476–484.

Soussi, T., Lozano, G., 2005. p53 mutation heterogeneity in cancer.

Biochem. Biophys. Res. Commun. 331, 834–842.

Sutherland, R., 1988. Cell and environment interactions in

tumor microregions: the multicell spheroid model. Science 240,

177–184.

Vohradsky, J., 2001. Neural model of the genetic network. J. Biol. Chem.

276, 36168–36173.

Walenta, S., Snyder, S., Haroon, Z., Braun, R., Amin, K., Brizel, D.,

Mueller-Klieser, W., Chance, B., Dewhirst, M., 2001. Tissue gradients

of energy metabolites mirror oxygen tension gradients in a rat

mammary carcinoma model. Int. J. Radiat. Oncol. Biol. Phys. 51,

840–848.

Warburg, O., 1930. The Metabolism of Tumors. Constable Press, London.

Yao, X., 1993. Review of evolutionary artificial networks. Int. J.

Intelligent Syst. 8, 539–567.

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An evolutionary hybrid cellular automaton model of solid tumour growth

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