I have been invited by Josep Vehi to give a talk about my research at the institute he directs at the University of Girona.
In this talk I have introduced for the first time the game theory model in which I have worked with people in Dresden (Andreas Deutsch, Haralambos Hatzikirou) and Bonn (Matthias Simon) in which I study Gatenby's hypothesis of acid mediated tumour invasion. I have also talked about the Cellular Automata model I have developed with Benjamin Ribba (Lyon) to study the microenvironmental influence in cancer evolution which I have previously presented in Lyon, Dundee and Dresden.
The presentation can be found here in PDF format.
Showing posts with label game theory. Show all posts
Showing posts with label game theory. Show all posts
Friday, June 01, 2007
Tuesday, April 03, 2007
Back from Scotland and Gatenby's talk
I am back from sunny Scotland in sunny Saxony. Of the remaining speakers in Dundee, the one whose talk I was looking for the most was the one from Robert Gatenby, Arizona University (as with Vito Quaranta, a life scientist).
I know the work of Gatenby because he is one of the few researchers involved in using evolutionary game theory (although not of the most conventional, fitness-and-payoff-table kind) to study cancer evolution. Specifically he is working on how acidity due to glycolysis (the anaerobic metabolism that constitutes and advantage for tumour cells that lack oxygen due to the distance to a blood vessel) is a necessary step in the evolution towards cancer. The so called Warburg effect is the result of a well known biochemical mechanism but, what is the evolutionary advantage?
As he has shown in other papers, the advantage for glycolytic cells is that the poison the environment of other cells so they face less competition. They also degrade the connective tissue and thus increase the motility of cells, which is a required step for a tumour to become invasive. From my point of view it is interesting that he seemed to imply that this acidification of the microenvironment is not only a facilitator for cancer but a necessary step. I guess that Hanahan and Weinberg could include this in the section for mechanisms for invasion and metastasis.
From the therapeutic point of view, his research suggests that either alkalising the microenvironment (to counteract the progressive acidification resulting from the glycolytic metabolism) or making it even more acidic by reducing the pH in the blood (and thus contributing to self poisoning of glycolytic cells) would be something worth trying.
I know the work of Gatenby because he is one of the few researchers involved in using evolutionary game theory (although not of the most conventional, fitness-and-payoff-table kind) to study cancer evolution. Specifically he is working on how acidity due to glycolysis (the anaerobic metabolism that constitutes and advantage for tumour cells that lack oxygen due to the distance to a blood vessel) is a necessary step in the evolution towards cancer. The so called Warburg effect is the result of a well known biochemical mechanism but, what is the evolutionary advantage?
As he has shown in other papers, the advantage for glycolytic cells is that the poison the environment of other cells so they face less competition. They also degrade the connective tissue and thus increase the motility of cells, which is a required step for a tumour to become invasive. From my point of view it is interesting that he seemed to imply that this acidification of the microenvironment is not only a facilitator for cancer but a necessary step. I guess that Hanahan and Weinberg could include this in the section for mechanisms for invasion and metastasis.
From the therapeutic point of view, his research suggests that either alkalising the microenvironment (to counteract the progressive acidification resulting from the glycolytic metabolism) or making it even more acidic by reducing the pH in the blood (and thus contributing to self poisoning of glycolytic cells) would be something worth trying.
Thursday, March 01, 2007
Old Dawkins video on cooperation
I have found this video on google video. It is an old Richard Dawkins BBC TV programme in which he goes through some topics of which I am quite fond and which I personally think have some yet-to-be-explored relevance to cancer research.
The documentary was produced a few years after Dawkins wrote The selfish gene and not long after Robert Axelrod had written The evolution of cooperation. It takes from Axelrod's research on cooperation (whose own take on how this can be observed in cancer has been mentioned in this blog before) to illustrate how cooperation might evolve in an place in which agents (say humans, bacteria or buffaloes) are selfish. The topics are the usual ones in game theory such as prisoners dilemma (how playing if for an undetermined number of times changes what is the best strategy), tragedy of the commons (if everybody is overusing a resource why shouldn't you, and if few people overuse, why shouldn't you since it will make no difference whatsoever?).
The tragedy of the commons seems to me a suitable game to model global warming (why should your country cut on carbon emissions if nobody else does or why should you if everybody does? this seems to apply to most countries but the likes of U.S. and China whose weight is to big to be considered just another player) or cancer (at the end, tumour cells do kill the host and thus themselves).
The documentary was produced a few years after Dawkins wrote The selfish gene and not long after Robert Axelrod had written The evolution of cooperation. It takes from Axelrod's research on cooperation (whose own take on how this can be observed in cancer has been mentioned in this blog before) to illustrate how cooperation might evolve in an place in which agents (say humans, bacteria or buffaloes) are selfish. The topics are the usual ones in game theory such as prisoners dilemma (how playing if for an undetermined number of times changes what is the best strategy), tragedy of the commons (if everybody is overusing a resource why shouldn't you, and if few people overuse, why shouldn't you since it will make no difference whatsoever?).
The tragedy of the commons seems to me a suitable game to model global warming (why should your country cut on carbon emissions if nobody else does or why should you if everybody does? this seems to apply to most countries but the likes of U.S. and China whose weight is to big to be considered just another player) or cancer (at the end, tumour cells do kill the host and thus themselves).
Friday, February 09, 2007
Axelrod et al: Evolution of cooperation among tumor cells
R. Axelrod, D. Axelrod and K. Pienta: Evolution of cooperation among tumor cells. PNAS vol 103, 36, pp. 13474-13479, 2006.
A few months ago a friend of mine from Vienna send me the link to this paper (thanks Peter!) and although I skimmed through it at the time only now did I have the chance to read it with a little bit more of care. Robert Axelrod is well known in the complex systems and game theory communities. The research he did almost a quarter of century ago (detailed in his book: The evolution of cooperation) explained how cooperation can be established between two agents (people, elephants or cells) even when the mechanisms of the cooperation have not been agreed beforehand and the agents could gain more in the short term by not cooperating.
Now Axelrod and coauthors speculate on how this approach could be used to study carcinogenesis. They present this in the framework of Hanahan and Weinberg and the six capabilities required to progress towards cancer (self sufficiency in growth signals, ignoring anti growth signals, evasion of apoptosis, angiogenesis, limitless replicative potential and invasion/metastasis).
Now, this paper is no regular paper. Most research papers I read describe a particular piece of clinical research (we have investigated this gene in this context...), mathematical or computational model (in this paper we introduce a model that explains the influence of acidity in...) or are review papers. This one does not describe new clinical research nor does propose a formal way to describe any aspect of oncology nor represents a review of carcinogenesis research from the cooperation point of view. This is not meant to be a criticism. The paper represents for me a new category of papers, one whose aim is not as much as telling finished research as to suggest to the reader new venues of research under a particular perspective.
If that was indeed the aim then this is a good paper. According to the authors, the conventional view on tumor progression using the Hanahan and Weinberg framework is that cells have to acquire all the six capabilities but under the new cooperation based view this is no longer necessary. It could be possible that, at least some of this capabilities are provided by some cells to others and thus cancer could occur when groups of cells displaying a mixed set of capabilities collaborate to create the same effect of a single cell acquiring all the capabilities and reaching fixation (taking over the tumour population) by clonal expansion. One of the things that I was not very comfortable with is that the authors state that cancers are the result of genetic (or epigenetic) instability. Readers of this site probably know that this is currently a hotly debated topic (something as fundamental such as: what starts carcinogenesis) and that in front of the Weinberg school (cancer starts from genetic instability) is the , say, Tomlison school (a bigger number of cells and selection suffices to explain the start of cancer). My view is that if tumour cells can cooperate in order to share capabilities and progress down the path of carcinogenesis then having a higher mutation rate might not be so relevant and thus a cooperation based view on cancer would favour the view that cancer does not really need genetic instability to get started. If this view of mine turns out to be a stupidity remember that you read it here first.
The paper provides a number of examples of capabilities in which cooperation can happen. In angiogenesis (where cells can produce growth factors that benefit not only the producing cell but others in the neighbourhood), self sufficiency from (certain) growth signals (there is a certain amount of growth signals which can be produced in paracrine or autocrine fashion) and in invasion/metastasis (collaboration to degrade the ExtraCellular Matrix).
The authors point out that this view of carcinogenesis arises a number of new research questions such as what are the resources that can be shared among cooperating tumour cells, what mechanisms are used to share these resources, how does this affect the order in which mutations appear (since mutations can appear in parallel)? Interesting questions but it might take some for someone to come with the answers...if it is that answers can be found using evolutionary cooperation.
A few months ago a friend of mine from Vienna send me the link to this paper (thanks Peter!) and although I skimmed through it at the time only now did I have the chance to read it with a little bit more of care. Robert Axelrod is well known in the complex systems and game theory communities. The research he did almost a quarter of century ago (detailed in his book: The evolution of cooperation) explained how cooperation can be established between two agents (people, elephants or cells) even when the mechanisms of the cooperation have not been agreed beforehand and the agents could gain more in the short term by not cooperating.
Now Axelrod and coauthors speculate on how this approach could be used to study carcinogenesis. They present this in the framework of Hanahan and Weinberg and the six capabilities required to progress towards cancer (self sufficiency in growth signals, ignoring anti growth signals, evasion of apoptosis, angiogenesis, limitless replicative potential and invasion/metastasis).
Now, this paper is no regular paper. Most research papers I read describe a particular piece of clinical research (we have investigated this gene in this context...), mathematical or computational model (in this paper we introduce a model that explains the influence of acidity in...) or are review papers. This one does not describe new clinical research nor does propose a formal way to describe any aspect of oncology nor represents a review of carcinogenesis research from the cooperation point of view. This is not meant to be a criticism. The paper represents for me a new category of papers, one whose aim is not as much as telling finished research as to suggest to the reader new venues of research under a particular perspective.
If that was indeed the aim then this is a good paper. According to the authors, the conventional view on tumor progression using the Hanahan and Weinberg framework is that cells have to acquire all the six capabilities but under the new cooperation based view this is no longer necessary. It could be possible that, at least some of this capabilities are provided by some cells to others and thus cancer could occur when groups of cells displaying a mixed set of capabilities collaborate to create the same effect of a single cell acquiring all the capabilities and reaching fixation (taking over the tumour population) by clonal expansion. One of the things that I was not very comfortable with is that the authors state that cancers are the result of genetic (or epigenetic) instability. Readers of this site probably know that this is currently a hotly debated topic (something as fundamental such as: what starts carcinogenesis) and that in front of the Weinberg school (cancer starts from genetic instability) is the , say, Tomlison school (a bigger number of cells and selection suffices to explain the start of cancer). My view is that if tumour cells can cooperate in order to share capabilities and progress down the path of carcinogenesis then having a higher mutation rate might not be so relevant and thus a cooperation based view on cancer would favour the view that cancer does not really need genetic instability to get started. If this view of mine turns out to be a stupidity remember that you read it here first.
The paper provides a number of examples of capabilities in which cooperation can happen. In angiogenesis (where cells can produce growth factors that benefit not only the producing cell but others in the neighbourhood), self sufficiency from (certain) growth signals (there is a certain amount of growth signals which can be produced in paracrine or autocrine fashion) and in invasion/metastasis (collaboration to degrade the ExtraCellular Matrix).
The authors point out that this view of carcinogenesis arises a number of new research questions such as what are the resources that can be shared among cooperating tumour cells, what mechanisms are used to share these resources, how does this affect the order in which mutations appear (since mutations can appear in parallel)? Interesting questions but it might take some for someone to come with the answers...if it is that answers can be found using evolutionary cooperation.
Tuesday, October 24, 2006
Bach et al: An evolutionary-game model of tumour-cell interactions
L.A. Bach, S.M. Bentzen, J. Alsner and F.B. Christiansen. An evolutionary-game model of tumour-cell interactions: possible relevance to gene therapy. European Journal of Cancer 27 (2001) 2116-2120.
Bach et al have taken the work from Tomlinson and Bodmer (which I reviewed a few day ago) in which angiogenesis is studied with the help a Game Theory. In the previous case the game involved two players who could chose to produce angiogenic factors or to do nothing. As long as one of the players is willing to shoulder on the cost of producing the factor, both players get the benefit. As expected the result is a polymorphism in which both types of strategies coexist. I said that this polymorphism is to be expected since if there were only factor producing cells then non cooperating cells will have an advantage (since they get the benefits without paying the costs) whereas if the population is made of non cooperating cells then factor producing cells will have an advantage (as long as the benefit of producing the factor is higher than the cost).
The revision of the model proposed by Bach et al considers the implications of extending the game to three players if the benefits of angiogenesis appear only if two out of the three players cooperate to produce the factor.
The payoff table would look like this:
A+,A+ A+,A- A-, A-
A+ 1-i+j 1-i+j 1-i
A- 1+j 1+j 1
where A+ means factor producing and A- means that is not factor producing. Also i is the cost of producing the factor and j is the benefit.
Bach et al analyse how these cost and benefit parameters affect the equilibria and the existence of polymorphism. For this they use computer simulations and find that when the benefit is three times the cost something interesting happens. In that case the final composition of the tumour population would be a consequence of the composition of the original population. Most likely this finding has limited (if any) consequences in potential therapies since physicians have not got the ability to change the initial composition of phenotypes in a tumour. Still, the (qualitative) results can be used as a guide for gene therapies.
Bach et al have taken the work from Tomlinson and Bodmer (which I reviewed a few day ago) in which angiogenesis is studied with the help a Game Theory. In the previous case the game involved two players who could chose to produce angiogenic factors or to do nothing. As long as one of the players is willing to shoulder on the cost of producing the factor, both players get the benefit. As expected the result is a polymorphism in which both types of strategies coexist. I said that this polymorphism is to be expected since if there were only factor producing cells then non cooperating cells will have an advantage (since they get the benefits without paying the costs) whereas if the population is made of non cooperating cells then factor producing cells will have an advantage (as long as the benefit of producing the factor is higher than the cost).
The revision of the model proposed by Bach et al considers the implications of extending the game to three players if the benefits of angiogenesis appear only if two out of the three players cooperate to produce the factor.
The payoff table would look like this:
A+,A+ A+,A- A-, A-
A+ 1-i+j 1-i+j 1-i
A- 1+j 1+j 1
where A+ means factor producing and A- means that is not factor producing. Also i is the cost of producing the factor and j is the benefit.
Bach et al analyse how these cost and benefit parameters affect the equilibria and the existence of polymorphism. For this they use computer simulations and find that when the benefit is three times the cost something interesting happens. In that case the final composition of the tumour population would be a consequence of the composition of the original population. Most likely this finding has limited (if any) consequences in potential therapies since physicians have not got the ability to change the initial composition of phenotypes in a tumour. Still, the (qualitative) results can be used as a guide for gene therapies.
Friday, October 13, 2006
Tomlinson and Bodmer: Modelling the consequences of interactions between tumour cells
I.P.M. Tomlinson and W.F. Bodmer. British Journal of Cancer 75 (2) 157-160, 1997.
Again Tomlinson, this time working with Bodmer, taking a couple of simple but interesting examples of the use of game theory for cancer research. This time they work on two different games: angiogenesis and apoptosis. In the first game angiogenic factors might be produced by tumour cells with the result that the cell producing and its neighbours reap the benefits of increased access to nutrients. In the second game cells might produce factors to escape angiogenesis which might or might not benefit the neighbours.
As it is usual in these cases the model assumes a large population of tumour cells, asexual reproduction and a population site that does not need to be constant since the object of the study is the frequency of particular phenotypes, not their absolute number. Further assumptions: the population of tumour cells is genetically diverse and this diversity is distributed homogeneously.
In the first game: angiogenesis, there are two strategies: either to produce or not to produce angiogenic factors. There is a cost associated to producing then and also a payoff. If you are a non producing tumour cell and you interact with a producing tumour cell you get the same benefits but none of the costs of a factor producing tumour cell. The result is that there are equilibria in which both phenotypes coexist as long as the cost of producing angiogenic factors is outweighed by the benefit.
The second game is more sophisticated. In this case we have three different strategies: either we produce factors to help neighbouring cells avoid apoptosis (paracrine factors), or we produce autocrine factors that help us avoid apoptosis or, alternatively, we might save ourselves all that trouble and do nothing. As usual there is a payoff table in which the different parameters represent the costs of producing paracrine and autocrine factors and the benefits they provide to whoever is the target of the factor. Tomlinson and Bodmer use GT to study different types of equilibria.
In this case it is easy to see that the first strategy is not viable since any group of cells doing nothing and reaping the benefits of endocrine factor producing cells would drive them to extintion. On the other hand if the cost of producing autocrine factors is smaller than the benefit of avoiding apoptosis then there will be a selection for cells capable of producing those autocrine factors.
The conclusion of the paper is that a tumour cell population might not adopt a strategy that would help the whole but does not confer any advantage to the individual that brings it (which is a reasonably safe proposition to make to people studying evolution). The question is again if there could be therapies designed to exploit the fact that tumour cells can stop cooperating among each other.
Again Tomlinson, this time working with Bodmer, taking a couple of simple but interesting examples of the use of game theory for cancer research. This time they work on two different games: angiogenesis and apoptosis. In the first game angiogenic factors might be produced by tumour cells with the result that the cell producing and its neighbours reap the benefits of increased access to nutrients. In the second game cells might produce factors to escape angiogenesis which might or might not benefit the neighbours.
As it is usual in these cases the model assumes a large population of tumour cells, asexual reproduction and a population site that does not need to be constant since the object of the study is the frequency of particular phenotypes, not their absolute number. Further assumptions: the population of tumour cells is genetically diverse and this diversity is distributed homogeneously.
In the first game: angiogenesis, there are two strategies: either to produce or not to produce angiogenic factors. There is a cost associated to producing then and also a payoff. If you are a non producing tumour cell and you interact with a producing tumour cell you get the same benefits but none of the costs of a factor producing tumour cell. The result is that there are equilibria in which both phenotypes coexist as long as the cost of producing angiogenic factors is outweighed by the benefit.
The second game is more sophisticated. In this case we have three different strategies: either we produce factors to help neighbouring cells avoid apoptosis (paracrine factors), or we produce autocrine factors that help us avoid apoptosis or, alternatively, we might save ourselves all that trouble and do nothing. As usual there is a payoff table in which the different parameters represent the costs of producing paracrine and autocrine factors and the benefits they provide to whoever is the target of the factor. Tomlinson and Bodmer use GT to study different types of equilibria.
In this case it is easy to see that the first strategy is not viable since any group of cells doing nothing and reaping the benefits of endocrine factor producing cells would drive them to extintion. On the other hand if the cost of producing autocrine factors is smaller than the benefit of avoiding apoptosis then there will be a selection for cells capable of producing those autocrine factors.
The conclusion of the paper is that a tumour cell population might not adopt a strategy that would help the whole but does not confer any advantage to the individual that brings it (which is a reasonably safe proposition to make to people studying evolution). The question is again if there could be therapies designed to exploit the fact that tumour cells can stop cooperating among each other.
Tuesday, October 10, 2006
I.P.M. Tomlinson: Game theory models of interactions between tumour cells
Review of EJC 35-9 (1997) 1495-1500.
This is the first paper I am aware of that uses game theory to study a problem in the field of cancer research. The paper is only from 1997 so it is easy to see that the field is ripe for further development.
The advantage of being the first to use a given tool in any area is that you can chose the problem and come with an simple and elegant study. Tomlinson has used a simple system in which tumour cells can adopt a number of strategies such as producing cytotoxic substances and cytotoxic resistance. The hypothesis is that some tumour cells attempt to gain advantage by actively harming neighbouring cells. This initial hypothesis is studied considering a number of different scenarios in which different phenotypes are combined. Initially there are three types of phenotypes or strategies: cells that can produce cytotoxic substances, cells that can produce factors that protect them from cytotoxic substances and finally cells that do none of this. Tomlinson presents a payoff table in which the interactions between the different phenotypes are presented in a parametrised way so he can hypothesise different values of the cost it represents to produce the toxin or the cost of producing the resistance or the benefit conferred to harm a player by doing so. Once the game is properly defined he goes on to study the potential equilibria by running simulations on a computer and varying the different parameters of the payoff table.
He also studies alternative strategies such as phenotypes that produce both the cytotoxic substance and its resistance and flexible strategies that behave differently according to the phenotype of the player they compete against. The conclusions he obtains are that several phenotypes can coexist simultaneously in a tumour (since there are configurations of parameters of the payoff table that lead to equilibrium), that flexible strategies are better than fixed ones (unless the cost of flexibility is too high and that therapies could be designed that could exploit the fact that tumour cells can harm other tumour cells under some circumstances (maybe promoting competition and not collaboration among them).
Of course the model is very very simple and the conclusions should be taken with some care (there are no spacial considerations, no hints of what the parameters of the payoff table could be, the results are not surprising). Still, this model gives some theoretical backing to these conclusions and suggests some ideas on how to design a therapy which is quite nice.
This is the first paper I am aware of that uses game theory to study a problem in the field of cancer research. The paper is only from 1997 so it is easy to see that the field is ripe for further development.
The advantage of being the first to use a given tool in any area is that you can chose the problem and come with an simple and elegant study. Tomlinson has used a simple system in which tumour cells can adopt a number of strategies such as producing cytotoxic substances and cytotoxic resistance. The hypothesis is that some tumour cells attempt to gain advantage by actively harming neighbouring cells. This initial hypothesis is studied considering a number of different scenarios in which different phenotypes are combined. Initially there are three types of phenotypes or strategies: cells that can produce cytotoxic substances, cells that can produce factors that protect them from cytotoxic substances and finally cells that do none of this. Tomlinson presents a payoff table in which the interactions between the different phenotypes are presented in a parametrised way so he can hypothesise different values of the cost it represents to produce the toxin or the cost of producing the resistance or the benefit conferred to harm a player by doing so. Once the game is properly defined he goes on to study the potential equilibria by running simulations on a computer and varying the different parameters of the payoff table.
He also studies alternative strategies such as phenotypes that produce both the cytotoxic substance and its resistance and flexible strategies that behave differently according to the phenotype of the player they compete against. The conclusions he obtains are that several phenotypes can coexist simultaneously in a tumour (since there are configurations of parameters of the payoff table that lead to equilibrium), that flexible strategies are better than fixed ones (unless the cost of flexibility is too high and that therapies could be designed that could exploit the fact that tumour cells can harm other tumour cells under some circumstances (maybe promoting competition and not collaboration among them).
Of course the model is very very simple and the conclusions should be taken with some care (there are no spacial considerations, no hints of what the parameters of the payoff table could be, the results are not surprising). Still, this model gives some theoretical backing to these conclusions and suggests some ideas on how to design a therapy which is quite nice.
Thursday, October 05, 2006
Mansury, Diggory and Deisboeck: Evolutionary game theory in an agent based brain tumor model: exploring the 'genotype-phernotype' link
Mansury et al. JTB 238 (2006) 146-156.
One of the things I had in mind when I started this blog is that I could use it to force me to write reviews about some of the most relevant papers that I often read for my own 'dirty' purposes. Usual reasons apply: it is good to write about what you read since synthesis helps understanding.
In any case, as you know, one of the topics I am interested on is cancer research using evolutionary game theory and although evolution is not what these people have studied the other important keywords are present in this paper.
Mansury et al have devised a nice spatial (2D lattice) agent based (Cellular Automata style) system in which tumour cells inhabit a space with nutrients. Tumour cells can be found in two varieties: A (proliferative) and B (migratory). Non evolutionary game theory is used to analyse the interactions between cells that have different phenotypes and how those interactions reflect on the payoffs of the individual cells and on the tumour as a whole. The payoffs in this game are slightly more complicated (and according to the authors, more realistic) than those of other games. The payoff of a cells is made of three different factors: communication payoff, proliferation payoff and migration payoff.
For the simulations (since it is quite difficult to come with a nice analytical study) they run CAs with 500x500 lattices in which nutrients are diffused from the centre and the middle. From here they study how changing the payoff table results in different velocity of tumour growth, different tumour surface roughness (useful to analyse the malignancy of a tumour) and the numbers of both tumour populations with time
From my point of view, the most significant shortcoming of an otherwise interesting piece of research (and acknowledged by the authors) is the lack of evolution in the model. With evolution out of the equation the condition under which phenotypes emerge and take over the original population cannot be studied. One of the nice features of game theory is that it can be used to study the equilibrium states of tumour cell populations when those tumours are studied as composed of individual cells (or agents in Mansury's et al model). Since the author's know this I am looking forward their next paper to see how the improved model can be used to study carcinogenesis.
One of the things I had in mind when I started this blog is that I could use it to force me to write reviews about some of the most relevant papers that I often read for my own 'dirty' purposes. Usual reasons apply: it is good to write about what you read since synthesis helps understanding.
In any case, as you know, one of the topics I am interested on is cancer research using evolutionary game theory and although evolution is not what these people have studied the other important keywords are present in this paper.
Mansury et al have devised a nice spatial (2D lattice) agent based (Cellular Automata style) system in which tumour cells inhabit a space with nutrients. Tumour cells can be found in two varieties: A (proliferative) and B (migratory). Non evolutionary game theory is used to analyse the interactions between cells that have different phenotypes and how those interactions reflect on the payoffs of the individual cells and on the tumour as a whole. The payoffs in this game are slightly more complicated (and according to the authors, more realistic) than those of other games. The payoff of a cells is made of three different factors: communication payoff, proliferation payoff and migration payoff.
For the simulations (since it is quite difficult to come with a nice analytical study) they run CAs with 500x500 lattices in which nutrients are diffused from the centre and the middle. From here they study how changing the payoff table results in different velocity of tumour growth, different tumour surface roughness (useful to analyse the malignancy of a tumour) and the numbers of both tumour populations with time
From my point of view, the most significant shortcoming of an otherwise interesting piece of research (and acknowledged by the authors) is the lack of evolution in the model. With evolution out of the equation the condition under which phenotypes emerge and take over the original population cannot be studied. One of the nice features of game theory is that it can be used to study the equilibrium states of tumour cell populations when those tumours are studied as composed of individual cells (or agents in Mansury's et al model). Since the author's know this I am looking forward their next paper to see how the improved model can be used to study carcinogenesis.
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