# Acidity and vascularization as linear goods in cancer

Last month, Robert Vander Velde discussed a striking similarity between the linear version of our model of two anti-correlated goods and the Hauert et al. (2002) optional public good game. Robert didn’t get a chance to go into the detailed math behind the scenes, so I wanted to do that today. The derivations here will be in the context of mathematical oncology, but will follow the earlier ecological work closely. There is only a small (and generally inconsequential) difference in the mathematics of the double anti-correlated goods and the optional public goods games. Keep your eye out for it, dear reader, and mention it in the comments if you catch it.[1]

In this post, I will remind you of the double goods game for acidity and vascularization, show you how to simplify the resulting fitness functions in the linear case — without using the approximations of the general case — and then classify the possible dynamics. From the classification of dynamics, I will speculate on how to treat the game to take us from one regime to another. In particular, we will see the importance of treating anemia, that buffer therapy can be effective, and not so much for bevacizumab.

### Strategies and factored replicator dynamics

Our model considers two goods:

1. Acidity produced by glycolysis is a public good for all tumour cells,[2] regardless of whether they have aerobic or anaerobic metabolism. A unit of acidity costs nothing to produce, except for the opportunity cost in not using oxygen, and provides a benefit of $b_a$ to be split among all. In contrast,
2. increased oxygen from vascularization is a club good that costs c to produce and that provides a benefit $b_v$ that is shared only among aerobic cells.

Since being a defector in the public good is required to benefit from the club good, we call the two goods anti-correlated. It also means that cooperating in both goods is always strictly worse than cooperating in acidity but defecting in vascularization, since neither you nor your kin can benefit from the vasculariture you invest energy in. Thus, we have three viable strategies:

• GLY — The glycolytic strategy is a cooperator in the public good since it does not use oxygen and thus produces acid. And a defector in the club good, since it does not (over) produce VEGF to call for more blood vessels and thus does not increase vascularization.
• VOP — This strategy is a defector in the public good since it uses oxygen and thus does not produce acid. But using oxygen means it can benefit from the club good, to which it contributes (cooperates) by (over)producing VEGF.
• DEF — This strategy is the universal defector, it uses oxygen and thus does not produce acid, and it also does not (over)produce VEGF.

In the classic style of public goods games,[3] we imagine the cells coming together at random into groups of n + 1 cells consisting of the focal agent and $n_G, n_V, n_D$ (sampled from the distributions $(x_G, x_V, x_D)$) many GLY, VOP, and DEF cells, respectively. This results in the fitness functions:

\begin{aligned} W_G & = \sum_{n_G + n_V + n_D = n} \binom{n}{n_G,n_V,n_D} x_G^{n_G} x_V^{n_V} x_D^{n_D} \frac{b_a (n_G + 1)}{n + 1} \\ & = \sum_{n_G = 1}^n x_G^{n_G}(1 - x_G)^{n - n_G} \frac{b_a (n_G + 1)}{n + 1} \\ W_V & = \sum_{n_G + n_V + n_D = n} \binom{n}{n_G,n_V,n_D} x_G^{n_G} x_V^{n_V} x_D^{n_D} ( \frac{b_a n_G}{n + 1} + \frac{b_v (n_V + 1)}{n - n_G + 1} - c ) \\ W_D & = \sum_{n_G + n_V + n_D = n} \binom{n}{n_G,n_V,n_D} x_G^{n_G} x_V^{n_V} x_D^{n_D} ( \frac{b_a n_G}{n + 1} + \frac{b_v n_V}{n - n_G + 1} ) \\ \end{aligned}

with the factored replicator dynamics of our system given by:

\begin{aligned} \dot{p} & = p(1 - p)(W_G - \langle W \rangle_{V,D}) \\ \dot{q} & = q(1 - q)(W_V - W_D) \end{aligned}

Where $p := x_G$ is the proportion of public good contributors, $q := x_V/(x_V + x_D)$ is the proportion of club good contributors, and $\langle W \rangle_{V,D} := qW_V + (1 - q)W_D$ is the average fitness of oxygen users.

### Simplifying the fitness functions

At first glance, the nonlinearity in the fitness functions might seem intimidating, but they can be greatly simplified by observing a simple combinatorial identity. Let us start with the q or vascularization gain function:

\begin{aligned} W_V - W_D & = \sum_{n_G + n_V + n_D = n} \binom{n}{n_G,n_V,n_D} x_G^{n_G} x_V^{n_V} x_D^{n_D} (\frac{b_v}{n - n_G + 1} - c) \\ & = \sum_{m = 0}^n \binom{n}{m} p^{n - m}(1 - p)^m \frac{b_v}{m + 1} - c \end{aligned}

where we relabeled by defining $m := n_V + n_D = n - n_G$ as the number of participants in the club good. The issue now is to eliminate the $m + 1$ in the denominator which can be done by observing that $\binom{n}{m} = \frac{m + 1}{n + 1}\binom{n + 1}{m + 1}$.

\begin{aligned} W_V - W_D & = \sum_{m = 0}^n \binom{n + 1}{m + 1} p^{n - m}(1 - p)^m \frac{b_v}{n + 1} - c \\ & = \frac{b_v}{(1 - p)(n + 1)}([\sum_{m' = 0}^{n'}\binom{n'}{m'}p^{n' - m'}(1 - p)^{m'}] - \binom{n'}{0}p^{n'}) - c\\ & = \frac{(1 - p^{n + 1}) b_v}{(1 - p)(n + 1)}- c, \end{aligned}

where in the intermittent step, I made the (re-)appearance of the sum of binomial distribution explicit by relabeling with n’ = n + 1 and m’ = m + 1.

We can repeat this for GLY vs. DEF to get:

\begin{aligned} W_G - W_D & = \sum_{n_G + n_V + n_D = n} \binom{n}{n_G,n_V,n_D} x_G^{n_G} x_V^{n_V} x_D^{n_D} (\frac{b_a}{n + 1} - \frac{b_v n_V}{n - n_G + 1}) \\ & = \frac{b_a}{n + 1} - \Big( \sum_{m = 0}^n \binom{n}{m} p^{n - m} (1 - p)^m \Big( \sum_{k = 0}^m \binom{m}{k} q^k(1 - q)^{m - k} \frac{b_v k }{m + 1} \Big) \Big) \\ & = \frac{b_a}{n + 1} - \sum_{m = 0}^n \binom{n}{m} p^{n - m} (1 - p)^m b_v q (1 - \frac{1}{m + 1}) \\ & = \frac{b_a}{n + 1} - b_v q + \frac{(1 - p^{n + 1}) q b_v}{(1 - p)(n + 1)}. \end{aligned}

And combine the above two results to give us the factored gain function for acidity:

\begin{aligned} W_G - \langle W \rangle_{V,D} & = W_G - (qW_V + (1 - q)W_D) \\ & = W_G - W_D - q(W_V - W_D) \\ & = \frac{b_a}{n + 1} - q b_v + \frac{q (1 - p^{n + 1})b_v }{(1 - p)(n + 1)} - q(\frac{(1 - p^{n + 1})b_v}{(1 - p)(n + 1)} - c) \\ & = \frac{b_a}{n + 1} - q(b_v - c). \end{aligned}

### Dynamic regimes and treatment order effect

From these gain functions, we can see that the the linear goods can have three possible dynamics:

1. If $b_a > (b_v - c)(n + 1)$ then the population converges towards all-GLY, else
2. if $b_a < (b_v - c)(n + 1)$ then one of two cases is possible:
1. If $b_v > c(n + 1)$ then the population converges towards all-VOP, else
2. if $b_v < c(n + 1)$ then the population will orbit around an internal fixed-point with $q^* = \frac{b_a}{(b_v - c)(n + 1)}$ and $1 - \frac{b_v}{c(n + 1)} \leq p^* \leq \frac{c(n + 1)}{b_v} - 1$.[4]

With the possible dynamic regimes in mind, we can think about treatment as changing the game parameters and taking us from one regime to another. Clinically, the worst outcome is a heterogeneous population (2.2) and the second worst is having a highly glycolytic tumour (1). Thus, we want treatment to move us towards dynamic regime (2.1) by having $b_v > c(n + 1)$ and $(b_v - c)(n + 1) > b_a$. This can be achieved either by increasing $b_v$ or decreasing $c$, or if the first condition is already satisfied then by decrease $b_a$.

Consistent with my discussion in March:

• angiogenesis inhibitors like bevacizumab look like a bad treatment in our model. Either increasing the cost c of producing vasculature or reducing its effectiveness (lowering $b_v$) is generally bad for the patient. In contrast,
• the ideal therapy in this model involves treating or avoiding anemia (so $b_v$ is high), and
• supplementing with buffer therapy (so decreasing $b_a$; for early empirical work, see Robey et al., 2009) if the first condition is met.

If we are starting with something close to all-VOP (dynamics 1 with $b_a > (b_v - c)(n + 1)$ and $b_v < c(n + 1)$) then the order of treatment also matters. Consider the example dynamics in the figure at right.[5] We see plots of the proportion of GLY (blue), VOP (magenta), DEF (gold) cells in the tumour versus time for two alternative treatment orders (between panels). In the top panel, at t = 1 we apply buffer therapy (normalize acidity) first and thus take the patient to regime 2.2 before we start treatment for anemia (normalize vascularization, t = 6), and thus push them towards a highly heterogeneous — and potentially dangerous — tumour in the interim. Instead, in the bottom panel if we first prep the patient by treating their anemia (normalize vascularization, t = 1), it might not have an apparent effect right away (if $b_a$ is very high, as it is in this example) but after we apply buffer therapy (normalize acidity, t = 6), the patient will be safely moved towards all-VOP without passing through a state with highly heterogeneous tumour.

Given that this is yet another limit of our general nonlinear anti-correlated goods model, it is nice to see the above agreement. Although this limit lacks nuance, it can be useful as a starting point: even non-linear acidity and vascularization functions will appear linear for small enough perturbations, allowing this limit model to act as a potential guide.

### Notes and References

1. Hint: think about the loneliest loner in Hauert et al. (2002).
2. By making acidity a benefit for all tumour cells, we are embracing the acid-mediated tumour invasion hypothesis (Gatenby 1995, Basanta et al., 2008). Of course, too much of a good thing can be bad, and it is possible to consider over-saturating functions like Archetti (2014), but I won’t deal with nonlinear functions like this one in this post.
3. It is possible to raise a number of reasonable objections to the public good as a model of cancer dynamics. However, I will shelve them for this discussion. Although since I don’t consider this approach pointless, I do have a more general case for defending it for a later post.
4. The exact position of $p^*$ is at the root of the equation $\frac{1 - p^n}{1 - p} = \frac{c(n + 1)}{b_v}$. This doesn’t have a simple closed form, but since we know that $\frac{1 - p^{n + 1}}{1 - p} = \sum_{k = 0}^n p^k$, we can bound the polynomial from below with $1 + p$ and from above with $\sum_{k = 0}^\infty p^k = \frac{1}{1 - p}$. From this follow the bounds on the root $p^*$.
5. If you want to experiment with the dynamics of treatment, just play with my Mathematica notebook dgLinB.np. Both panels have n = 2, c = 1, and start with $b_v = 1.1$, $b_a = 10$. After buffer therapy is initiated, $b_a$ drops to -1, and after curing anemia, $b_v$ rises to 3.1. The only difference between the panels is which time step they get the therapies.

Archetti, M. (2014). Evolutionary dynamics of the Warburg effect: glycolysis as a collective action problem among cancer cells. Journal of Theoretical Biology, 341: 1-8

Basanta, D., Simon, M., Hatzikirou, H., & Deutsch, A. (2008). Evolutionary game theory elucidates the role of glycolysis in glioma progression and invasion. Cell Proliferation, 41(6): 980-987.

Gatenby, R. A. (1995). The potential role of transformation-induced metabolic changes in tumor-host interaction. Cancer Research, 55(18): 4151-4156.

Hauert, C., De Monte, S., Hofbauer, J., & Sigmund, K. (2002). Replicator dynamics for optional public good games. Journal of Theoretical Biology, 218 (2), 187-94 PMID: 12381291

Robey, I. F., Baggett, B. K., Kirkpatrick, N. D., Roe, D. J., Dosescu, J., Sloane, B. F., … & Gillies, R. J. (2009). Bicarbonate increases tumor pH and inhibits spontaneous metastases. Cancer Research, 69(6): 2260-2268.