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A Game Theoretical Perspective on the Somatic Evolution of Cancer

Environmental and genetic mutations can transform the cells in a co-operating healthy tissue into an ecosystem of individualistic tumour cells that compete for space and resources. Various selection forces are responsible for driving the evolution of cells in a tumour towards more malignant and aggressive phenotypes that tend to have a fitness advantage over the older populations. Although the evolutionary nature of cancer has been recognised for more than three decades (ever since the seminal work of Nowell) it has been only recently that tools traditionally used by ecological and evolutionary researchers have been adopted to study the evolution of cancer phenotypes in populations of individuals capable of co-operation and competition. In this chapter we will describe game theory as an important tool to study the emergence of cell phenotypes in a tumour and will critically review some of its applications in cancer research. These applications demonstrate that game theory can be used to understand the dynamics of somatic cancer evolution and suggest new therapies in which this knowledge could be applied to gain some control over the evolution of the tumour.


Introduction
Environmental and genetic mutations can transform the cells in a co-operating healthy tissue into an ecosystem of individualistic tumour cells that compete for space and resources [1,2,3]. If we consider a tumour as an ecosystem it is possible to utilise tools traditionally used by ecologists to study the evolution of a population in which there is some degree of phenotypical diversity. One such tool is evolutionary game theory (EGT) which merges traditional game theory with population biology [4]. It allows the prediction of successful phenotypes and their adaptation to environmental selection forces. EGT is considered as a promising tool in which to frame oncological problems [5] and has been recently made more relevant by phenotypic studies of carcinogenesis such as the ones by Hanahan, Weinberg and colleagues [6,7].
Game theory (GT) was introduced by von Neumann and Morgenstern as an instrument to study human behaviour [8,9]. A game describes the interactions of two or more players that follow two or more well defined strategies in which the benefit of each player (payoff) results from these interactions [10]. GT can be employed to study situations in which several players make decisions in order to maximise their own benefit. GT was initially introduced to model problems in economics, social and behavioural sciences and is used as a formal way to analyse interactions between agents that behave strategically. Evolutionary game theory is the application of conventional GT as used by economists and sociologists to study evolution and population ecology [4]. As opposed to conventional GT, in EGT the behaviour of the players is not assumed to be based on rational payoff maximisation but it is thought to have been shaped by trial and error -adaptation through natural selection or individual learning [11]. In the context of the evolution of populations there are two GT concepts that have to be interpreted in a different light. First, a strategy is not a deliberate course of action but a phenotypic trait. The payoff is Darwinian fitness, that is, average reproductive success. Secondly, 2 David Basanta*, Andreas Deutsch the players are members of a population that compete or cooperate to obtain a larger share of the population [4].
To illustrate some of the ideas in EGT let us consider the following example named the Hawk-Dove game [11]. In this game we study an imaginary population of individuals and a resource V which affects the reproductive success of the individuals in this population. The population contains two phenotypes that represent two different strategies to access the resource. When two individuals compete for the resource the outcome will depend on the phenotypic strategies involved. The first phenotype, called Hawk in the game, always escalates the fight until injured (at a cost in fitness equal to C) or until the rival retreats. The second phenotype, known as Dove in the game, will retreat if the opponent escalates, that is, if the opponent seems determined to fight. The interactions between the different phenotypes are shown in the payoff table 1.1.  Table 1.1 presents the interactions between the different phenotypes considered in the game. The table should be read following the columns such that the payoff for a Hawk playing another Hawk is V −C 2 expressing the fact that they both have to share the resource and that they stand an equal chance of getting injured. The payoff of a Hawk playing a Dove is V since the Dove will withdraw from the competition. A Dove playing a Hawk gets no payoff since it withdraws and when playing another Dove it will get the resource V in half of the occasions. With this information it is possible to predict that if the population is mainly composed of individuals with the Dove phenotype then a Hawk individual will have a significant fitness advantage (as in most of the interactions, the rivals are likely to be a Dove and thus retreat from a full scale fight for the resource). On the other hand if the fitness cost of injury is more than twice as high as the benefit provided by the contested resource then a Dove would be quite successful in a population dominated by Hawks (since a Hawk that interacts frequently with other Hawks is likely to be eventually wounded and a Dove will always avoid costly wounds). Using this example Maynard Smith introduced the concept of an evolutionary stable strategy (ESS) [11]. An ESS is defined as a phenotype that, if adopted by the vast majority of a population, will not be displaced by any other phenotype that could appear in the population as a result of evolution [11]. Under this definition the Dove phenotype cannot be an ESS and only under some specific circumstances (when the fitness benefit of getting the resource outweighs the fitness cost of an injury) would a Hawk phenotype be evolutionary stable.
GT has been used to address many problems in biology in which different species or phenotypes within one species compete. Examples of this are the evolution of sex ratios [12], the emergence of animal communication [13] and fighting behaviour and territoriality [11]. A recent focus on the capabilities that cells acquire as tumours evolve [6] has shown how the interplay between different phenotypes with different capabilities can lead to different evolutionary paths. This stresses the importance of GT as a modelling tool in cancer. The most important capabilities which cells have to acquire in a neoplasm that will become a malignant tumour are shown in figure 1.1. They include: unlimited replicative potential, environmental independence for growth, evasion of apoptosis, angiogenesis and invasion. The circumstances in which these capabilities evolve and spread through the tumour population can be studied using GT. Some of the most important milestones in the transformation of a healthy tissue cells into malignant cancer such as tumourigenesis, angiogenesis and invasion, have already been approached with GT.

Cell 58
Acquired GS autonomy was the first of the six capabilities to be clearly defined by cancer researchers, in large part because of the prevalence of dominant oncogenes that have been found to modulate it. Three common molecular strategies for achieving autonomy are evident, involving alteration of extracellular growth signals, of transcellular transducers of those signals, or of intracellular circuits that translate those signals into action. While most soluble mitogenic growth factors (GFs) are made by one cell type in order to stimulate proliferation of another-the process of heterotypic signaling-many cancer cells acquire the ability to synthesize GFs to which they are responsive, creating a positive feedback signaling loop often termed autocrine stimulation (Fedi et al., 1997). Clearly, the manufacture of a GF by a cancer cell obviates dependence on GFs from other cells within the tissue. The production of PDGF (platelet-derived growth factor) and TGF␣ (tumor growth factor ␣) by glioblastomas and sarcomas, respectively, are two illustrative examples (Fedi et al., 1997).
The cell surface receptors that transduce growthstimulatory signals into the cell interior are themselves targets of deregulation during tumor pathogenesis. GF receptors, often carrying tyrosine kinase activities in their cytoplasmic domains, are overexpressed in many cancers. Receptor overexpression may enable the cancer cell to become hyperresponsive to ambient levels (EGF-R/erbB) is upregulated in stomach, brain, and breast tumors, while the HER2/neu receptor is overexpressed in stomach and mammary carcinomas (Slamon et al., 1987; Yarden and Ullrich, 1988). Additionally, gross We describe each capability in turn below, illustrate with overexpression of GF receptors can elicit ligand-indea few examples its functional importance, and indicate pendent signaling (DiFiore et al., 1987). Ligand-indepenstrategies by which it is acquired in human cancers. dent signaling can also be achieved through structural alteration of receptors; for example, truncated versions Acquired Capability: Self-Sufficiency of the EGF receptor lacking much of its cytoplasmic in Growth Signals domain fire constitutively (Fedi et al., 1997). Normal cells require mitogenic growth signals (GS) be-Cancer cells can also switch the types of extracellular fore they can move from a quiescent state into an active matrix receptors (integrins) they express, favoring ones proliferative state. These signals are transmitted into the that transmit progrowth signals (Lukashev and Werb, cell by transmembrane receptors that bind distinctive 1998; Giancotti and Ruoslahti, 1999). These bifunctional, classes of signaling molecules: diffusible growth facheterodimeric cell surface receptors physically link cells tors, extracellular matrix components, and cell-to-cell to extracellular superstructures known as the extracelluadhesion/interaction molecules. To our knowledge, no lar matrix (ECM). Successful binding to specific moieties type of normal cell can proliferate in the absence of of the ECM enables the integrin receptors to transduce such stimulatory signals. Many of the oncogenes in the signals into the cytoplasm that influence cell behavior, cancer catalog act by mimicking normal growth signalranging from quiescence in normal tissue to motility, ing in one way or another.
resistance to apoptosis, and entrance into the active Dependence on growth signaling is apparent when cell cycle. Conversely, the failure of integrins to forge propagating normal cells in culture, which typically prothese extracellular links can impair cell motility, induce liferate only when supplied with appropriate diffusible apoptosis, or cause cell cycle arrest (Giancotti and Rumitogenic factors and a proper substratum for their inteoslahti, 1999). Both ligand-activated GF receptors and grins. Such behavior contrasts strongly with that of tuprogrowth integrins engaged to extracellular matrix mor cells, which invariably show a greatly reduced components can activate the SOS-Ras-Raf-MAP kinase dependence on exogenous growth stimulation. The conpathway (Aplin et al., 1998; Giancotti and Ruoslahti, clusion is that tumor cells generate many of their own 1999). growth signals, thereby reducing their dependence on The most complex mechanisms of acquired GS autonstimulation from their normal tissue microenvironment.
omy derive from alterations in components of the down-This liberation from dependence on exogenously destream cytoplasmic circuitry that receives and prorived signals disrupts a critically important homeostatic cesses the signals emitted by ligand-activated GF mechanism that normally operates to ensure a proper receptors and integrins. The SOS-Ras-Raf-MAPK cascade plays a central role here. In about 25% of human behavior of the various cell types within a tissue. if not all cancers have acquired the same set of capabilities during their evolution. These capabilities are: unlimited replicative potential, environmental independence from anti growth signals, production of own growth signals, evasion of apoptosis, angiogenesis and invasion (from [6] with permission).
The remaining of this chapter will provide, to the best of our knowledge, all the relevant examples of the application of GT to the study of the somatic cancer evolution and finally hint some of the possible future venues of this method in the context of cancer research.

Tumourigenesis
Tumour initiation requires the acquisition of a number of phenotypic capabilities such as evasion of apoptosis and independence from environmental signals (see figure 1.1). The evolution of these capabilities, normally acquired when the tumour is still in the avascular stage, are studied in research by Tomlinson and Bodmer [14,15] and by Gatenby and Vincent [16].

Evasion of apoptosis
Problem. Apoptosis or programmed cell death is a mechanism that hinders tumour progression. Cells with a working apoptotic machinery die when genetic abnormalities are detected [6]. Thus cells in a malignant cancer have to evolve mechanisms to disable the apoptotic machinery.
Model. Tomlinson and Bodmer [14] present a model in which three different apoptosis evasion related strategies are considered: 1. Cells that produce a paracrine growth factor to prevent apoptosis of neighbouring cells. 2. Cells that produce an autocrine growth factor to prevent apoptosis of themselves. 3. Cells susceptible to paracrine growth factors but incapable of production of factors.
The aim of the model is to study the possibility of stable coexistence of the different phenotypes (polymorphism) that could be possible in a tumour when only these three phenotypes are considered. Parameter a represents the cost of producing a paracrine factor, b is the cost of the factor produced in an autocrine fashion, c is the fitness benefit of evading apoptosis.
In table 1.2, a is the cost of producing the paracrine factor, b the benefit of receiving the paracrine factor and c the benefit of producing the autocrine factor.
Results. If a is positive then the third strategy displaces the first one from the population. If only the other two strategies are considered then if the benefit of the autocrine factor, provided by c, is positive the second strategy will displace the third one. In most relevant situations the model shows a strong selection for the autocrine factor producing phenotype and under the 1 A game theoretical perspective on the somatic evolution of cancer 5 assumptions of the model the altruistic strategy (the first one) will always be displaced.
Remarks. The model is very simple and easy to understand while at the same time it captures the relevant features necessary to study the evolution of the mechanisms to avoid apoptosis. However the authors do not explain the mechanisms by which these phenotypes could appear in a tumour and what would be the biological explanation of the costs and benefits of the different growth factors.

Environmental poisoning
Problem. Tomlison introduced a further model in which he considers the hypothesis that tumour cells might boost their own replicative potential at the expense of other tumour cells by evolving the capability of producing cytotoxic substances [15].
Model. Tomlinson speculates with different strategies that cells may adopt to produce or cope with toxic factors. The main model aims to study the polymorphic equilibria when cells can adopt one of the three following strategies: 1. Cells producing cytotoxic substances against other cells, 2. cells producing resistance to external cytotoxic substances, and 3. cells producing neither cytotoxins nor resistance. Payoff table for the change in fitness for cells in a tumour in which the base payoff is z, e the cost of producing the cytotoxin, f the fitness cost of being affected by the cytotoxin, g the advantage of subjecting another cell to the cytotoin and finally the cost of developing resistance to the cytotoxin is h.
Results. Game theoretical analysis and simulations show that production of cytotoxic substances against other tumour cells can evolve in a tumour population and that several cytotoxin related strategies may be present at a given time (polymorphism).
Remarks. Although the author admits that there is little experimental evidence for mutations that cause tumour cells to harm their neighbours, the Warburg effect could fit nicely in the framework presented in this work. The Warburg effect describes the switch of tumour cells from the conventional aerobic metabolism to the glycolytic metabolism. This metabolism is less efficient but produces, as a by-product, acid that can harm neighbouring cells [17,18,19,20]. Thus, in the game described by Tomlinson, e could correspond to the fitness loss of the less efficient glycolytic metabolism, f could be the fitness loss of a normal cell in an acid environment and g the fitness benefit received by glycolytic cells that can take advantage of the harm done to their non glycolytic neighbours.

Angiogenesis
Problem. A very important capability that has to be acquired by tumours on the path to cancer is angiogenesis. Without access to the circulatory system tumours do not grow to sizes bigger than 2mm in diameter [21]. Cells capable of angiogenesis produce growth factors that promote the creation of new blood vessels that can provide nutrients and oxygen to previously unreachable areas in a growing tumour. Presumably the factors will be produced at a cost to the tumour cell.
Model I. In their interpretation of an angiogenic game Tomlinson and Bodmer [14] consider two strategies: cells denoted as A+ can produce angiogenic factors at a fitness cost i and cells denoted as A-that produce no angiogenic factors. In any case cells will get a benefit j when there is an interaction involving an angiogenic factor producing cell. The payoffs for the interactions between these cells are shown in table 1.4. Table 1.4. Payoff table for the change in fitness for a cell in a tumour with cells capable of producing angiogenic factors (A+) and cells susceptible to benefit from growth factors (A-).
Results. The model shows that as long as the benefit j of angiogenesis is greater than the cost i of producing angiogenic factors then both types of strategies will be present in a tumour in proportion to these costs.
Remarks. This model of angiogenesis is rather simplistic but constitutes a nice foundation for later models that take into account spatial considerations. A more significant drawback is that the model does not attempt to suggest a link between the different fitness costs and benefits and the underlying biological mechanisms.
Model II. The first extension to this model was proposed by Bach et al [22] suggesting a game with interactions between three players. In this game the benefit j of angiogenesis is obtained only when at least two of the three 1 A game theoretical perspective on the somatic evolution of cancer 7 players produce the angiogenic factor. The new payoff table is shown in table 1.5. Table 1.5. Payoff table for the change in fitness for a cell in a tumour with cells capable of producing angiogenic factors (A+) and cells susceptible to benefit from growth factors (A-). In this version the interactions involve three players. The payoff of a player is given by the columns (ie. the payoff of a A+ cell interacting with a A+ and a A-cell is 1 -i + j but a A+ interacting with two A-cells is 1 -j).
Results. The authors produce simulations using table 1.5 and show (see figure 1.2) that this game yields results than differ from the original ones introduced by Tomlinson and Bodmer. In this case even when the cost i is smaller than the benefit j there are many scenarios for which the angiogenic strategy will be displaced from the population. The existence of a polymorphic equilibrium containing the angiogenic strategy depends, intuitively, on the cost of producing angiogenic factors in comparison to the benefit they give but also on the relative frequency of cells playing the angiogenic strategy in the population. These results suggest that a gene therapy against the reparation of tumour supressor genes would only need to change a fraction of the mutated cells before the dynamics of the system drives them to extinction.
Model III. In a separate research Bach et al [23] studied how a spatial version of the angiogenesis game could produce different results to those of the original model by Tomlinson and Bodmer. In this game players inhabit a 100x100 lattice. Each player can follow either the angiogenic (A+) or the nonangiogenic (A-) strategy. Time is discrete and in each time step a number of cells is removed from the lattice at random. Neighbouring cells compete (using table 1.4) to occupy unallocated space. The candidate cell that achieves the highest score interacting with the neighbours determines the strategy that will be followed by the new cell in the vacant slot (see figure 1.3).
Results. The authors found the results of the new formulation of the model markedly different from those of the non-spatial counterpart. In spatial models, space tends to favour growth promoters in ways that cannot be seen in the non-spatial model. In any case polymorphic equilibria do exist and in both spatial and non spatial cases the proportion of angiogenic players increases as the benefits of angiogenesis increase or as the cost of producing the growth factors decreases. The authors also conclude that, contrary to other evolutionary models [24], space does not significantly favour co-operative strategies in populations of cells.
¼ 1=2: a stable equilibrium exists above v ¼ 1=2 and unstable equilibrium below v ¼ 1=2. The two internal uilibria will change their positions symmetrically as a nction of the relationship between the cost and benefit rameters. For j/i tending to infinity the two equilibria nds toward frequencies of A+ cell going to 1 or 0. Fig. 1 shows a simulation with j=3i. The frequency of + cells converges towards the stable internal equiliium or towards zero depending on the initial relative equency of these cells. The unstable equilibrium acts a barrier that has to be crossed, in order to change e frequency of A+ cells away from either of the two able equilibria. The population will reach one of the o equilibria depending on whether the initial freency of A+ cells is located above or below the stable equilibrium. In other words, starting above e unstable equilibrium the population frequency stribution will end up in a stable clonic heteroneity (in population genetics termed a stable polyorphism).
change the fraction of mutated tumour cells below a certain level, and that once the frequency gets below the unstable equilibrium, internal dynamics will force the mutated cells to extinction. In cancer gene therapy, a great deal of effort has gone into restoring normal p53 function in mutated tumour cells [16]. If immunohistochemical detection of p53 is accepted as an indication of mutations, hotspots of mutated cells have been described [17]. Within these regions, a varying frequency of ble 2 tness matrix for the threshold model  The ordinate is the frequency of A+ cells and the abscissa is the number of cell generations. (from [22] with permission ) equation for proportion of growth promoting cells is then The equilibrium proportion of growth promoters in the case where b . c and d . b þ e 2 c is It appears, therefore, that the additional benefit, e does not change the general conclusion that the production of costly angiogenic factors is selected for amongst the cells. However, the equilibrium number of growth promoting cells is increased as the extra benefit increases (Maynard- Smith, 1982).
Other models of cell interactions are encompassed by the model in Table II. For example, Tomlinson and Bodmer (1997) also consider competition between cells which produce a factor to prevent programmed cell death in a purely paracrine fashion (i.e. no effect on producer) against cells which produce growth factor in a purely autocrine fashion (i.e. benefit to self but not to neighbours). By defining b 2 c , 0 the autocrine strategy now represents a purely "altruistic" act on the part of the growth promoter, as d and e remain constant. We see that T . 1 but S , 0; giving the well-known Prisoner's Dilemma game. Here, the interesting result is that although these "altruistic" cells would thrive in a pure population of growth promoters, in the well-mixed case, an invading non-promoting cell would reproduce and take over the entire population. This is seen by finding an equilibrium for the replicator equation (3) when c . b: In this case, the proportion of growth promoting cells is x Ã ¼ 0 corresponding to the defecting or non-promoting strategy being an evolutionarily stable strategy (Hofbauer and Sigmund, 1988).

SPATIAL EVOLUTIONARY GAME MODELS
Spatial patterns undoubtedly arise in the growth of cancerous tumours, for the simple reason that offspring cells grow adjacent to the parent cell. It is important, therefore, that spatial effects are accounted for when attempting to determine the growth and changing that for many games, space has an important role in determining the equilibrium proportion of defectors and co-operators. In general, discrepancies are almost always found between results from the mean-field replicator equation and nearest neighbour interaction models.
We now propose a variety of simple two dimensional lattice simulation models to determine the behaviour of such cell interaction dynamics (Wolfram, 1984;Nowak and May, 1992). Each simulation model begins with a 100 £ 100 array of cells, a proportion of which are growth promoting (C strategy) and some of which are nonpromoting (D strategy). The basic protocol for updating the cells on each generation is as follows: First, a certain number of cells will be removed from the lattice (the number depends on whether updating is synchronous, semi-synchronous or asynchronous, see below for details). The neighbours of the removed cells will then compete to occupy the empty cells. Each non-empty neighbour to the empty cell will perform a two-player interaction with each of its neighbours and hence receive a payoff. The payoff to reach interaction is determined according to the specified payoff matrix (in our case we use payoff Tables I and II). The obtained payoffs from each interaction are summed to yield the neighbour cell's total payoff or fitness, which in turn is used in order to determine which of the candidate neighbour cells will be selected to reproduce (the reproduction may be deterministic where the cell with the highest fitness reproduces or probabilistic where reproduction is proportional to fitness, again see below for details). The "winning" neighbour cell is subsequently copied into the empty site. Hence, an empty site always becomes re-occupied by the proliferation of the cell which wins a local competition. Figure 1 gives a diagrammatic representation of the update scheme for a single removed cell.   [23]. First a cell is selected to be removed (in this case the C marked in bold in the configuration on the left). Then the four neighbouring cells all assess their fitness (middle configuration). The cell with the highest fitness then reproduces into the empty space (right). (from [23] with permission)

Motility/invasion
A tumour in which cells develop the capability of invading other tissues becomes a malignant tumour and thus significantly worsens the prognosis of a patient. EGT is a tool that could greatly improve our understanding of the circumstances that influence the successful evolution of invasive phenotypes.
Model I. Mansury and colleagues have recently introduced a rather unconventional EGT formulation that they used on top of a previous cellular automaton model in the context of brain tumours [25,26]. This GT module allows the original model to deal with cell-cell interactions in a tumour consisting of cells with different phenotypes. The model was designed to investigate the genotype to phenotype link in a polymorphic tumour cell population.
This model covers two distinct strategies. Strategy P is characterised by a highly proliferative genotype and a high number of gap junctions. Strategy M is characterised by a highly migratory phenotype and a low number of gap junctions (which are used for cell to cell communication). As opposed to other game theoretical models in which a payoff table determines the fitness change of players when they interact, in this model three tables are used not to compute fitness change but to encode the rules of the cellular automata. The tables are shown in tables 1.6, 1.7 and 1.8.  Table 1.6 shows how the cell to cell communication capabilities change according to the strategies followed by the interacting cells. Since the authors assume that there is a negative correlation between the extent of communication and proliferation then the proliferative capability of a cell with the proliferative strategy P will be lower if it interacts with another cell with the same strategy. On the other hand it will be higher if it interacts with a cell with the motile strategy M. Table 1.7 shows that cells will have a higher proliferative potential when they interact with cells that have the motile strategy. Table 1.8 shows that interacting with cells with the motile strategy will also increase the probability of motility of the interacting cell. These tables are used to guide the behaviour of the cells on a 500x500 lattice containing initially 5 cells of each type and two unequal sources of nutrients in two different locations. In each time step the interactions of a cell with its neighbours are used to adjust the probabilities of proliferation and motility.
Results.The authors used this model to study how varying the payoffs for A-A interactions affects a number of features of the tumour such as the speed (see figure 1.4) at which it reaches the nutrient sources or the fractality of the resulting spatial patterns. liferate or migrate automatically enter a reversible, quiescent state.

Results
In the following, we show results from varying the payoffs for A-A cell-cell interactions in terms of their phenotypic proliferative activities (see Table 4). Recall from our modeling section that interactions between a pair of A-A cells result in the steepest reduction of the proliferation activity for both cells involved. We have also experimented with various migration payoffs for B-B interactions, yet as expected we did not find a discernible pattern in their spatio-temporal performance due to our current setup of equal time-scale for both proliferation and migration, which implies that during either proliferation or migration, the tumor expands to a vacant lattice site that is one unit of distance away from the parent's location for the former or the migrating cell's original location for the latter.

Average velocity
Here, average velocity serves as a measure for the tumor's overall performance. Fig. 1 shows that as on the x-axis the payoffs conferred to A-A interaction increase in terms of lesser reduction of Type A's proliferative activities, unexpectedly the average velocity of the tumor's spatial expansion exhibits a phase transition. That is, initially the time to the second nutrient source rises, representing declining velocity, as the A-A payoffs increase. But then it appears to reach a minimum velocity at a payoff of approximately 0.83 before it rises again, as Type A cells become the numerically dominating genotype (see the right y-axis in Fig. 2) in the total population (i.e. including proliferating, migrating, and quiescent cancer cells) at a payoff of 1.00. Fig. 2 shows that as the payoffs conferred to A-A cell-cell interaction increase, surface roughness declines continuously (left y-axis). At the same time, the frequency of genotype A (right y-axis) in the total population increases monotonically, i.e. becomes more robust. That is, as the more proliferative yet less migratory Type A becomes dominant, surface roughness declines. Conversely, surface roughness rises as the more migratory Type B becomes the dominating genotype (at lower A-A payoffs), consistent with our previous study examining 'structural-pattern' relationship (Mansury and Deisboeck, 2004) showing a positive correlation between fractal dimensions of the tumor surface and the number of migrating cells.

Time series and spatial profile of tumor genotypes
To better understand the evolution of the tumor cells over time, here we examine the time series of Remarks. Although the model is interesting it is not as simple as other models reviewed in this chapter. A significant drawback is that the model is not evolutionary in the sense that it does not take into account the possibility of mutations introducing new phenotypes or tumour cells producing offspring different from their progenitors. This has been identified by the authors as something to be addressed in a future version of the model although they still claim the model to be based on EGT. Moreover, the authors do not take advantage of the tools provided by GT analysis to study the different steady state situations that could arise for different values of the payoff tables. This is probably due to the fact that the model is not a conventional GT model and the conventional game theoretical tools would not be easy to use in this context. Also, the model provides a view on the dynamics of the tumour growth that is rarely found in most other game theory analysis whose focus is on the study of equilibria.
Model II. The emergence of invasive phenotypes is influenced not only by its interaction with one phenotype or the other but by the complex interplay of several phenotypes which, in many cases has an indirect effect. Basanta et al. [27] hypothesise a number of scenarios in which three types of phenotypes interact in games with two and three players. They place these phenotypes in the context of an evolutionary non-spatial game theoretical model to test the hypothesis by Gatenby and colleagues that tumour invasion is promoted by the emergence of cells with a glycolytic metabolism [20]. The model represents a glioma tumour populated by cells with enhanced proliferative capabilities, known as autonomous growth cells (AG), which can mutate into cells whose phenotype can make then follow either a more motile strategy (INV) or the glycolytic metabolism strategy (GLY). Table 1.9. Payoff table that represents the change in fitness of a tumour cell with a given phenotype interacting with another cell. Three different strategies are considered, those with higher replicative potential (AG), enhanced motility (INV) and glycolytic metabolism (GLY). The base payoff in a given interaction is equal to 1 and the cost of moving to another location with respect to the base payoff is c. The fitness cost of acidity is n and k is the fitness cost of having a less efficient glycolytic metabolism. The table should be read following the columns, thus the fitness change for an INV cell interacting with an AG would be 1 − c.

AG INV GLY AG
Results. Table 1.9 defines the interactions between the three phenotypes. The authors investigate a number of scenarios in which subgames based on two strategies are used to study how one phenotype could emerge in a tumour populated by cells that use a different strategy. The results show that cells with a higher replicative potential (AG) and invasive cells (INV) can coexist in a tumour as long as the fitness cost of motility is not too high. They also reveal that autonomous growth (AG) cells cannot coexist with glycolytic (GLY) cells. An interesting result of the model when all three strategies are considered simultaneously is that the appearance of the invasive phenotype is facilitated by the existence of glycolytic cells. Figure 1.6 shows how the proportion of invasive cells (X axis) increases as the cost of having a glycolytic metabolism (k) decreases and the cost of living in an acid environment (n) increases. In other words, the success of the invasive phenotype depends on the same factors that determine the fate of the glycolytic phenotype. The model suggests that any therapy that could increase the fitness cost of tumour cells switching to a glycolytic metabolism or the susceptibility of normal cells to acid environments might decrease the probability of the emergence of more invasive phenotypes. Fig. 1.5. Proportion of invasive cells in a tumour with three phenotypes (autonomous growth, invasive and glycolytic). k is the cost in terms of fitness of adopting the glycolytic metabolism whereas n is the fitness cost of a normal cell when staying with a glycolytic cell.

Outlook
The examples described in this chapter treat tumour populations as ecosystems of potentially co-operating and/or competing cells. In these ecosystems, the success of one phenotype depends on its interactions with other existing phenotypes. Such an approach has been shown to be a helpful and useful way to study cancer evolution. Most of the applications use GT to study the steady state of a population of tumour cells that follow different strategies dictated by their phenotypes, acquired as a result of genetic and epigenetic mutations. This can be very relevant to study the different ways in which a tumour population may evolve under different model parameters and assumptions of their interplay. Such studies could lead to cancer therapies that would alter the dynamics of cancer evolution towards benign tumours. One limitation of EGT/GT is that all relevant potential phenotypes/strategies have to be known a priori if a good understanding of cancer evolution is to be obtained. Knowing all the potential relevant phenotypes might be difficult and even if the phenotypes and their interactions are well known the EGT 1 A game theoretical perspective on the somatic evolution of cancer 13 model might be too complicated to be analysed. Moreover, with more strategies/phenotypes the composition of the population may not converge to an equilibrium and the frequencies of the phenotypes could keep oscillating in a regular or chaotic fashion [4].
One more limitation of the GT models shown in this chapter is that they do not study the dynamics in a tumour population (dynamics which may or may not lead to an equilibrium). GT models that make use of population biology have the potential to overcome this limitation and also ease the connection between a quantitative model and experimental data as the payoff tables used in more conventional EGT models tend to make the assumption that the fitness values are independent of space or time. One promising venue is to couple conventional GT with population dynamics which is also based on the assumption that successful strategies spread [28,4]. This trend is shown in the work of Gatenby and Vincent [16] whose EGT model uses methods from population biology in order to study how the phenotypes of cells in a population evolve towards ESS. Gatenby and Vincent adopted a game theory approach influenced by population dynamics to study the influence of the tumour-host interface in colorectal carcinogenesis. The authors formulated an extended system of Lotka-Volterra equations to model the effect of nutrients and sensitivity to growth constraints in the proliferation of tumour cells. The cells in the tumour population are characterised by the number of substrate transporters (a higher number of them allow more nutrients in the cell) on the cell surface and by the cell response to normal growth constraints. Initially all the tumour cells are assumed to have normal values for both parameters but these values are allowed to change and the authors study their evolution. Their work demonstrates that normal cells in a multicellular organism occupy a ridge-shaped maximum of the fitness landscape that allows the heterogeneous coexistance of multiple cells. This fact makes them susceptible to mutations that are fitter and thus allow the somatic evolution that characterises cancer. The authors also conclude that any therapy that reduces the population density would be counterproductive since it would allow the more aggressive phenotypes to grow. Subsequent research uses the same formulation to study other stages of tumour progression like invasion and metastasis [29,30].
GT is also a suitable tool to frame cooperative effects. Evolution of cooperation is a sub-field of GT pioneered by Robert Axelrod [31]. Researchers in this field study the circumstances under which selfish agents will spontaneously co-operate. An example of how evolution of cooperation could be used to study cancer evolution was provided by Axelrod and collaborators in a recent paper [32] in which they show how tumour cells can cooperate sharing skills and capabilities (such as the production of angiogenic factors or paracrine growth factors needed to escape the homeostatic regulation of the tissue).
Evolutionary game theory has a relatively short history in the field of theoretical oncology but with an increasingly better understanding of the role fitness of any cell using a specific strategy is determined by the point on the landscape corresponding to the two components of the strategy vector. A possible initial state for a pluripotent normal cell is indicated by the dot. This strategy corresponds to evolutionary equilibrium for normal cells and remains a maximum point for every snapshot as the landscape changes shape underneath it. Because of the interactive, nonlinear dynamics, the characteristics of the growing cell population alter the adaptive landscape. In the case of normal tissue, the differentiated cell populations ultimately find themselves at a point on a ridge-shaped maximum.
Interestingly, this novel configuration allows stable coexistence of multiple cellular "species" defined by the same G-function but with different "strategies" along the edge of the ridge. These results provide a simple, general control mechanism for simultaneous direction of pluripotent stem cells into a population with a specific differentiated phenotype and maintenance of a society of stable, noncompeting cellular populations in functioning tissue, a necessary condition for formation of multicellular organisms. This general mechanism is supported by experimental data demonstrating that differentiation in embryogenesis and tissue repair is controlled by signals from the microenvironment (18 -21).
Note the orientation of the ridge in nonequilibrium tissue (at t ϭ 0) along the v 2 axis, which defines the cellular interaction with normal environmental proliferation constraints. This indicates that coexistent, noncompeting populations must possess different strategies defined by these normal tissue controls (i.e., proliferation constraints by different combinations of oncogenes and tumor suppressor genes). That is, organization of function in developing or remodeling normal tissue (i.e., tissue not at a stable steady state) requires neighboring cells to possess different sets of growth control parameters (i.e., different "strategies"). The requirement for distinctly different growth control strategies in coexisting cellular populations provides a potential explanation for the large number of oncogenes and tumor suppressor genes found in the human genome and the diversity of their mutations found among cancers of different cell types.
It appears that invasive cancer is the "price" of the fitness landscape that allows formation of multicellular organisms. Because normal cells do not achieve a proper maximum at equilibrium, there is an opportunity for tissue invasion by mutant cellular phenotypes. As long as the mutants are derived by the same G-function, they do not change the shape of the adaptive landscape, and they will simply coexist at low numbers. However, if a mutant should have a different G-function, the situation is quite different (22). In fact, because the system dynamics are highly nonlinear, the introduction of a mutant population initially at a higher fitness, as discussed below, can deform the adaptive landscape such that normal cells now occupy a local minimum. This situation is illustrated in Fig. 2, where it is shown how the presence of a tumor cell deforms the normal cell adaptive landscape so that, at equilibrium, the normal cells are at a local minimum on the adaptive landscape. The tumor cells are introduced in small numbers (x 2 ϭ 10 at t ϭ 0) and grow to x 2 ϭ 218 at equilibrium. Because neither the normal or tumor cells are allowed to evolve, the tumor remains at a this equilibrium value and does not affect the number of normal cells. However, because the normal cells are now at a minimum, the stability of normal tissue requires proliferative controls that are both redundant (i.e., several oncogenes and tumor suppressor genes) and robust (i.e., maintenance of a sufficiently low basal rate of cellular mutations to prevent significant cellular evolution during the reproductive life of the organism).
We have seen that normal tissue may be invaded if the mutant population can change the adaptive landscape in a detrimental way (to normal cells). We will now show that if the mutant cells can evolve (toward a peak on the mutant adaptive landscape), the combined effect allows for the mutant type to replace the normal cells. We propose the transition of cellular populations from their normal configuration on the adaptive landscape to one that promotes emergence of a mutant phenotype constitutes the somatic evolution of an invasive cancer.
Evolution of Invasive Cancer. The model demonstrates that the evolutionary steps of invasive cancer take place at bifurcations in the system dynamics that result in a change of the equilibrium state. We now examine this process using the evolution of colorectal cancer as a specific example. Because we find that each normal cell population must be controlled by different sets of oncogenes and tumor suppressor genes, favored mutations in specific genetic loci will vary from one cell type to another. In colorectal carcinogenesis these targets appear to include, among others, the APC and K-RAS genes (3,23,24). The former is a multifunctional gene that acts as a tumor suppressor through down-regulation of ␤-catenin, which is a transcription activator (25) found at intercellular junctions (from which it transits to the nucleus), suggesting a role in transmitting information regarding contact with neighboring cells and other aspects of the microenvironment. The majority of K-RAS mutations are gain-of-function missense leading to increasing growth stimulatory signals, although they also affect cell adhesion, cell cycle, and cell metabolism (26).
Thus, K-RAS or APC mutations will diminish environmental control of proliferation in the tumor cells through increased translation of growth promoter signals and decreased inhibition from cell-cell contact (27). This is incorporated into the model by increasing the mean tissue carrying capacity of tumor cells.
With a normal background mutation rate of about 2 ϫ 10 Ϫ7 mutations/ gene/cell division (28), n ϭ t ϭ 0.0001. Mutations in u will be very close to the mean value, resulting in a very small rate of evolution. This is accounted for in the figures by setting SD of mutations among the normal and tumor cells to a small value. In generating Fig. 2, all parameter values for the normal cells are the same as those used in Fig. 1, and all parameter values for the tumor Fig. 1. Upon reaching equilibrium, normal cells actually sit on ridge-shaped maximum in the in vivo adaptive landscape. This provides a tissue structure that allows coexistence of multiple distinct but noncompeting populations along the length of the ridge, a necessary condition for formation of multicellular organisms. strategy corresponds to evolutionary equilibrium for normal cells and remains a maximum point for every snapshot as the landscape changes shape underneath it. Because of the interactive, nonlinear dynamics, the characteristics of the growing cell population alter the adaptive landscape. In the case of normal tissue, the differentiated cell populations ultimately find themselves at a point on a ridge-shaped maximum.
Interestingly, this novel configuration allows stable coexistence of multiple cellular "species" defined by the same G-function but with different "strategies" along the edge of the ridge. These results provide a simple, general control mechanism for simultaneous direction of pluripotent stem cells into a population with a specific differentiated phenotype and maintenance of a society of stable, noncompeting cellular populations in functioning tissue, a necessary condition for formation of multicellular organisms. This general mechanism is supported by experimental data demonstrating that differentiation in embryogenesis and tissue repair is controlled by signals from the microenvironment (18 -21).
Note the orientation of the ridge in nonequilibrium tissue (at t ϭ 0) along the v 2 axis, which defines the cellular interaction with normal environmental proliferation constraints. This indicates that coexistent, noncompeting populations must possess different strategies defined by these normal tissue controls (i.e., proliferation constraints by different combinations of oncogenes and tumor suppressor genes). That is, organization of function in developing or remodeling normal tissue (i.e., tissue not at a stable steady state) requires neighboring cells to possess different sets of growth control parameters (i.e., different "strategies"). The requirement for distinctly different growth control strategies in coexisting cellular populations provides a potential explanation for the large number of oncogenes and tumor suppressor genes found in the human genome and the diversity of their mutations found among cancers of different cell types.
It appears that invasive cancer is the "price" of the fitness landscape that allows formation of multicellular organisms. Because normal cells do not achieve a proper maximum at equilibrium, there is an opportunity for tissue invasion by mutant cellular phenotypes. As long as the mutants are derived by the same G-function, they do not change the shape of the adaptive landscape, and they will simply coexist at low numbers. However, if a mutant should have a different G-function, the situation is quite different (22). In fact, because the system dynamics are highly nonlinear, the introduction of a mutant population initially at a higher fitness, as discussed below, can deform the adaptive landscape such that normal cells now occupy a local minimum. This situation is illustrated in Fig. 2, where it is shown how the presence of a tumor cell deforms the normal cell adaptive landscape so that, at equilibrium, the normal gesting a role in transmitting information regarding contact with neighboring cells and other aspects of the microenvironment. The majority of K-RAS mutations are gain-of-function missense leading to increasing growth stimulatory signals, although they also affect cell adhesion, cell cycle, and cell metabolism (26).
Thus, K-RAS or APC mutations will diminish environmental control of proliferation in the tumor cells through increased translation of growth promoter signals and decreased inhibition from cell-cell contact (27). This is incorporated into the model by increasing the mean tissue carrying capacity of tumor cells.
With a normal background mutation rate of about 2 ϫ 10 Ϫ7 mutations/ gene/cell division (28), n ϭ t ϭ 0.0001. Mutations in u will be very close to the mean value, resulting in a very small rate of evolution. This is accounted for in the figures by setting SD of mutations among the normal and tumor cells to a small value. In generating Fig. 2, all parameter values for the normal cells are the same as those used in Fig. 1, and all parameter values for the tumor Fig. 2. The presence of tumor cells deforms the tissue adaptive landscape over time so that normal cells arrive at a local minimum and are then readily subject to invasion by the mutant phenotypes.

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(b) Fig. 1.6. (a) Upon reaching equilibrium, normal cells sit on the ridge-shaped maximum in the in vivo adaptive landscape. This provides a tissue structure that allows coexistence of multiple distinct noncompeting populations, a necessary condition for the formation of multicellular organisms. (b) The presence of tumour cells deforms the tissue adaptive landscape over time so that normal cells arrive at a local minimum and are then readly sunject to invasion by the mutant phenotypes (from [16] with permission).
of the microenvironment in tumour evolution [33] and with the recent interest in studying cancer from ecological [2,3] and phenotypic [6] viewpoints, the role of this tool to understand the interactions between all the relevant agents within a tumour and their role driving cancer evolution will surely rise.