# Game theoretic analysis of motility in cancer metastasis

As I am starting to write this post, the last of the 4th of July fireworks are winding down outside. I am currently traveling through New York on my way to Swarmfest 2013 in Orlando, FL and to visit David Basanta and Jacob Scott at the Integrated Mathematical Oncology Department of Moffitt Cancer Research Institute in Tampa, FL. The weather is unbearably hot already, so I imagine I will struggle in Florida, especially since I forgot all my shorts in Montreal!

The more important struggle, however, will be my lack of background in biology and medicine. On Jake’s suggestion, I decided to look at a paper of David’s on an evolutionary game theoretic approach to the emergence of motility in cancer cells. As far as I understand, one of the key steps in going from a benign tumor to a malignant cancer is metastasis or the ability of a cancer to spread from one organ to another non-adjacent organ. To achieve this, a cancer cell has to transition from a simple proliferative cell to a motile one. However, motility usually involves a cost to the organism.

Basanta, Hatzikirou, & Deutsch (2008) represent this cost by the payoff matrix:

$\begin{pmatrix} \frac{b}{2} + \frac{1}{2}(b - c) & b - c \\ b & \frac{b}{2} \end{pmatrix}$

Where player 1 is Motile and player 2 is Proliferative. Note that in the paper, there is a typo (they write 1/2 instead of b/2) and a different convention for writing payoff matrices (I write the payoffs for the row player, they write them for the column player). To understand the interaction, think of two cells meeting at random in a resource spot. If both are motile then one of them gets to stay in the resource spot and consume all the resources, and the other has to pay a cost $c$ to move and find a new empty site with resources $b$. Which case happens to a particular cell is by chance, so they are weighted by $1/2$. On the other hand, if a motile cell meets a proliferate cell then the motile cell will have to move for sure ($b - c$) but the proliferate can stay and eat all the resources ($b$). Finally, if two proliferate cells are in the same resource spot then they simply share the resources $b/2$ for each.

It is tempting to object to motile cells being able to always find an empty resource site, but it is not a difficult objection to overcome. Suppose that after a move, a motile cell only finds a new empty spot with probability $r$, then the post move payoff is $rb - c$. However, that can just as easily be captured by adjusting the cost to be $c' = c + (1 - r)b$ while not introducing in extra parameter. In fact, the game only has one parameter. Intuitively: what units are $b$ and $c$ measured in? Does it matter, or is it only their ratio that matters? If we write the game in the standard cooperate-defect game format, then it becomes (assuming $b > c$):

$\begin{pmatrix} 1 & 1 - \frac{c}{b - c} \\ 1 + \frac{c}{b - c} & 0 \end{pmatrix}$

If $c \geq b/2$ then the game is a Prisoner’s dilemma (and so motiles do not persist in the population), else it is a Hawk-Dove game with a stable polyclonal proportion of $p = 1 - \frac{c}{b - c}$ motile cells.

Of course, in a real organism, cells are not distributed randomly, but have some spatial structure. Basanta et al. (2008) consider this case with a simple square lattice model. Unfortunately, their model cannot be compared directly to the replicator dynamics treatment. Only qualitative comparison is possible and it suggests the same sort of results: for low ‘cost’ (it is parametrized a little bit more vaguely in the agent-based model) there is a polymorphism of motile and proliferate cells. As the ‘cost’ increases, the proportion of motiles decreases. I am a little bit skeptical of the ABM because it doesn’t run until stability (only 5000 cycles long on a 1000-by-1000 grid), but instead tries to look at the transient dynamics when there is new spots available to colonize. This makes sense to study in the context on cancer spread, however, a square lattice fails at allowing the most standard feature of cancer: exponential growth. On a square lattice, the population growth cannot be faster than quadratic and I doubt that is representative of biology.

Instead, I would consider a network structure that has fast mixing times and thus allows for exponential growth. One candidate would be to rewire randomly some of the grid edges to allow for short-distance hops, but that would still confine us to simulation. If we want to have an analytic treatment that we can compare directly to the inviscid model then I would opt for the Ohtsuki-Nowak transform on k-regular random graphs; this approach was also suggested by Jake over email. The ON transform allows us to incorporate this simple spatial structure into a modified payoff matrix and then just consider standard replicator dynamics of this modified game. For motile-vs-proliferate, the modified game for the local (and more cooperation inducing) Death-Birth dynamics is:

$\begin{pmatrix} 1 & 1 - \frac{c}{b - c} + \frac{1}{k - 2}(1 - \frac{2}{k + 1}\frac{c}{b - c}) \\ 1 + \frac{c}{b - c} - \frac{1}{k - 2}(1 - \frac{2}{k + 1}\frac{c}{b - c}) & 0 \end{pmatrix}$

Where $k$ is the degree of the graph. If $k + 1 \geq \frac{2}{b - c}$ then this game shifts favor towards motile agents, allowing a greater number of them to persist in polymorphic populations and increasing the cost for which the game shifts to PD dynamics. More importantly, if $(k - 1 + \frac{1}{k + 1})c \geq b$. In other words, if $c$ and $b$ are fixed, then the max degree for which we can have a stable all motile population is $b/c + 1/3 < k_{\max} < b/c + 1$

Unfortunately, this analysis is also limited. Although it would be nice to imagine that the agents are moving around the graph, that is simply not how the transform is built. Instead, we have to think of each edge as an abstract ‘area’, and a motile cell with some edges incident means that those areas have motile cells. Inside each area, there are several spots to share and the motile cells can move between these spots while proliferate cells cannot. As such, this misses the main point of metastasis. The cells are not moving from one organ to another, they are just competing in a local environment structured into areas. Finally, the population size is held fixed in this analysis, so it misses the primary justification for itself. Although motile cells can spread exponentially quickly through the population of motile and proliferate cells, the overall population is kept static. We can consider workarounds by suggesting that the total population is proportional to the average fitness in the original game, and so as the spatial structure promotes cooperation and the population grows. However, it might be just easier to instead consider a proper treatment of this as an ecological game.

Basanta, D., Hatzikirou, H., & Deutsch, A. (2008). Studying the emergence of invasiveness in tumours using game theory. The European Physical Journal B, 63 (3), 393-397 DOI: 10.1140/epjb/e2008-00249-y