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Don’t treat the player, treat the game: buffer therapy and bevacizumab

March 26, 2016 by Artem Kaznatcheev 9 Comments

No matter how much I like modeling for the sake of modeling, or science for the sake of science, working in a hospital adds some constraints. At some point people look over at you measuring games in the Petri dish and ask “why are you doing this?” They expect an answer that involves something that benefits patients. That might mean prevention, early detection, or minimizing side-effects. But in most cases it means treatment: how does your work help us treat cancer? Here, I think, evolutionary game theory — and the Darwinian view of cancer more generally — offers a useful insight in the titular slogan: don’t treat the player, treat the game.

One of the most salient negative features of cancer is the tumour — the abnormal mass of cancer cells. It seems natural to concentrate on getting rid of these cells, or at least reducing their numbers. This is why tumour volume has become a popular surrogate endpoint for clinical trials. This is treating the player. Instead, evolutionary medicine would ask us to find the conditions that caused the system to evolve towards the state of having a large tumour and how we can change those conditions. Evolutionary therapy aims to change the environmental pressures on the tumour, such that the cancerous phenotypes are no longer favoured and are driven to extinction (or kept in check) by Darwinian forces. The goal is to change the game so that cancer proves to be a non-viable strategy.^[1]

In this post I want to look at the pairwise game version of my joint work with Robert Vander Velde, David Basanta, and Jacob Scott on the Warburg effect (Warburg, 1956; Gatenby & Gillies, 2004) and acid-mediated tumour invasion (Gatenby, 1995; Gatenby & Gawlinski, 2003). Since in this work we are concerned with the effects of acidity and vascularization on cancer dynamics, I will concentrate on interventions that affect acidity (buffer therapy; for early empirical work, see Robey et al., 2009) or vascularization (angiogenesis inhibitor therapy like bevacizumab).

My goal isn’t to say something new about these therapies, but to use them as illustrations for the importance of changing between qualitatively different dynamic regimes. In particular, I will be dealing with the oncological equivalent of a spherical cow in frictionless vacuum. I have tried to add some caveats in the footnotes, but these could be multiplied indefinitely without reaching an acceptably complete picture.

Pairwise game for Warburg effect

Let’s start by looking at the game and updating terminology from the earlier post. We consider three strategies that differ in their relationships to vascularization and acidity:

GLY — The glycolytic strategy does not use oxygen and thus produces acid. It does not (over) produce VEGF to call for more blood vessels and thus does not increase vascularization.
VOP — This strategy uses oxygen and thus does not produce acid. It also (over) produces VEGF and thus calls for more blood vessels, directly increasing vascularization.
DEF — This strategy is the universal defector, it uses oxygen and thus does not produce acid, and it also does not (over)produce VEGF.

This results in the following payoff matrix for interactions between the three strategies:

$\begin{pmatrix} a_2 & a_1 & a_1 \\ a_1 + v_1 - c & a_0 + v_2 - c & a_0 + v_1 - c \\ a_1 + v_0 & a_0 + v_1 & a_0 + v_0 \end{pmatrix}$

Where $\{a,v\}_{0,1,2}$ corresponds to fitness effects of low, medium, and high acidity or vascularization (respectively), and c is the cost of VEGF (over)production. Note that the cost of acid-production (i.e. glycolysis) is captured implicitly by the lack of benefit to GLY from the oxygen of vascularization.

To conserve notation, let $\Delta a_0 := a_1 - a_0$ be the change in benefit of acidity going from low to medium and similarly for v and other subscripts. We suppose that the fitness functions for acid and vascularization are concave, so $\Delta a_0 \geq \delta a_1$ and $\Delta v_0 \geq \Delta v_1$ . The interesting dynamics are restricted to the parameter ranges of $\Delta v_1 < c < \Delta v_0$ and $\Delta a_0 > v_1 - c > v_0$ and $\Delta a_1 < v_2 - c < v_1$ . Outside of this range, at least one of the strategies is strictly dominated by at least one other strategy and interesting three-strategy dynamics disappear.

Depending on the parameter values, we have 9 qualitatively different dynamic regimes, represented below:

The 9 dynamic regimes of the acid-mediated invasion game.

The 9 possible dynamic regimes for the acid-mediated tumour invasion game.

In the above figure, we can imagine moving between the dynamic regimes by either changing the value of $\Delta a_0$ (the x-axis), $\Delta a_1$ (the y-axis), or by moving the borders between the regimes by changing any of the $v_{0,1,2}$ or c. Note that $q^* = \frac{\Delta v_0 - c}{\Delta v_0 - \Delta v_1}$ is the equilibrium proportion of VOP on the VOP–DEF edge.

Of course, before trying to move between dynamic regimes, it is important to consider which outcomes are desirable and which are undesirable. The worst possible result is a highly heterogeneous tumour, so we want to avoid an attracting internal fixed point — basins of attractions for this are marked in red in the above figure (two dynamic regimes). The second worst result is to have any other population with a lot of GLY cells, because those — or at least the acid they produce — are associated with more invasive phenotypes. The basins of attraction for these are marked in yellow (for all-GLY; 3 dynamic regimes) or orange (mix of GLY–VOP; 1 dynamic regime). Note that in three of the parameter regions, the dynamics have two distinct basins of attraction and thus different initial proportions can lead to different final proportions.^[2] In the other six regions, all three-strategy initial conditions lead to the same equilibrium point.

Acidity and buffer therapy

An example curve of fitness versus acidity. In blue is the original game, and in red and green are the first and second buffer therapies considered in this post.

The last ingredient we need for looking at treatment is a more explicit form for the fitness versus acidity and fitness versus vascularization curves and how these are affected by treatment.

Consider the figure on the left of fitness versus acidity.^[3] We can think of $x_0$ as the base level of acidity — when there is no or very few GLY cells in the local microenvironment, $x_1$ as the acidity correspond to a few GLY cells, and $x_2$ as the acidity corresponding to many GLY cells. Their mappings to $a_{0,1,2}$ are shown in blue in the figure at left. There are two ways that we can imagine buffer therapy working. One way is to lower the overall acidity of the environment and thus shifting all three of $x_{0,1,2}$ to the left by the same amount. This is shown in red in the figure at left. Since we are considering saturating curves with negative second derivatives, this means that both of $\Delta a_{0,1}$ would increase. A second way is to lower the amount of acidity produced by each GLY cell, that would decrease $\Delta x_{0,1}$ but leave $x_0$ unchanged. This is shown in green in the figure at left. With this therapy, $\Delta a_0$ would decrease and $\Delta a_1$ could decrease or increase depending on the strength of the therapy and the particulars of the fitness-acidity curve. In the particular case of the figure at left, $\Delta a_1$ remains about the same or increases every so slightly.

An ideal therapy for our system would increase $a_0$ and decrease $a_1$ , so neither of the above two buffer therapies would be effective in all cases. If we have the dynamics leading to high heterogeneity (i.e. if $\Delta a_0 < v_0 + q^* \Delta v_0$ and $\Delta a_1 < v_1 - c$ ) then lowering the overall acidity of the environment is preferable. If we have the dynamics in the orange basin (or the yellow basin above it; so $\Delta a_0 > v_2 - c$ and $\Delta a_1 > v_0$ ) then lowering the amount of acid produced by each GLY cell is preferable.^[4]

Vascularization and bevacizumab

For vascularization, we can consider a similar graph of vascularization to fitness as we did for acidity. But to talk about treatment, it is useful to consider the complications that come with a common side-effect of both cancer and cancer treatment — anemia. The blood of somebody suffering from anemia has a lowered ability to carry oxygen. This means that the same amount of vascularization provides less benefit for aerobic cells — both $v_{1,2}$ decrease, and $v_0$ remains about the same. In other words, anemia has the same effect on vascularization as our second buffer therapy had on acidity. However, vascularization and acidity play opposite roles in the dynamics of our system. So anemia is almost always bad for the patient. In particular, it increases the chance of ending up in one of the regimes with the yellow or orange basins of attraction and tightens the range of parameters for favourable dynamics. This means that a first treatment aimed at vascularization is to avoid or reverse anemia. There are numerous approaches to do this — from healthier diets to blood transfusions to blood doping with erythropoietin (usually with drugs like epoetin or darbepoetin^[5]) — and picking which approach is the best depends on factors outside our model.

Due to the larger parameter space (each of $v_0, v_1, v_2, c$ matters), it is a bit more difficult to characterize this model’s ideal therapy for vascularization. It is tempting to consider the strongly saturating regime where $v_1 = v_2$ , but then three of the dynamic regimes (two of them desirable) disappear because we get $v_0 + q^*\Delta v_0 = v_1 - c = v_2 - c$ . This leaves us with only one desirable dynamic regime — given by $\Delta a_0 > v_1 - c$ and $\Delta a_1 < v_0$ — and therapies leading to it need to increase $v_0, c$ and decrease $v_1 = v_2$ . If $v_1 < v_2$ then a different sort of ideal therapy becomes possible, one that increases both of $v_{1,2}$ but with a larger increase for $v_2$ than for $v_1$ . Unfortunately, given the saturating nature of benefit for producing enzymes and growth factors like VEGF (Hemker & Hemker, 1969; Archetti, 2013; Archetti et al., 2015), it seems unlikely that such a treatment is possible.

So what does this mean for bevacizumab and other angiogenesis inhibitor? Hard to say. Bevacizumab is believed to bind to extracellular VEGF, stopping it from acting as a messenger between normal and tumour cells and surrounding blood-vessels, and thus inhibiting the production of more blood vessels (Ranieri et al., 2006). In our model, this seems like a decrease in $v_{1,2}$ while keeping constant $v_0, c$ ; given the saturating nature of the vascularization benefit curve, we would also expect the decrease in $v_1$ to be greater than the decrease in $v_2$ . This is similar to the effects of anemia on the benefit of oxygen, and of our second buffer therapy on acidity. As I discussed earlier, anemia is generally bad for the patient in this model, and so I would seldom expect to see benefit from bevacizumab to treating this game.^[6]

Notes and References

Of course, traditional therapies can also be viewed from the evolutionary perspective. Killing cancer cells would just be the introduction of a particular evolutionary pressure that doesn’t favor the cells. But by broadening to a perspective where you always think explicitly in terms of evolutionary pressures, you open your eyes to a wider range of therapies that are less direct in the way they deal with cancer.
With two basins of attractions, it is possible to transform traditional ‘treat the player’ therapies into evolutionary ones. If the population is in an undesired basin of attraction — one that leads to an undesired equilibrium point — but still close to the border between the basins then we could target one of the strategies to push the population into the other basin. Once in the other basin, we can stop treatment and let evolution take its route, and move the population towards the desired equilibrium point.
The non-linearity of the fitness versus acidity/vascularization curves show my debt to Archetti (2013, 2014). As I discussed in the first post on this 3-strategy game, it can be viewed as the n = 1 of our coupled public and club goods game. This work combines Archetti (2013) as a model of growth factors — the club good — and Archetti (2014) as a model of acidity — the public good, to show results that are not possible in the two separate models. However, I think the 3-strategy game presented here can also stand on its own, without being seen as the limit of public goods game. It can be agnostic to the exact implementation of the hand-wavy relationship between the fitnesses of (sub)populations of tumour cells. In the end, if we wanted to parametrize this model then we will likely be restricted to measuring only these population level relationships, and so it is better to remain agnostic on the individual-level story in something like a public-good game.
There are also important factors to consider outside the model. For example, if we believe the acid-mediated tumour invasion hypothesis (Gatenby, 1995; Gatenby & Gawlinski, 2003; or for treatment considerations see Robey et al., 2009) then reducing overall acidity might lead to fewer metastases. Even if we don’t believe in acid-mediated tumour invasion, studies like Raghunand et al. (1999) suggest that increasing tumour pH (i.e. decreasing acidity) can help traditional chemotherapies — like doxorubicin for breast cancer — to be more effective.
Increasing erythropoietin is a good way to highlight the importance of unintended side-effects. Although combating anemia is seen as a positive, using blood doping can increase cancer invasiveness (Mohyeldin et al., 2005) and death through blood clots (Bennett et al., 2006) in patients. Neither of these aspects are foreseeable from the model that I presented in this post.
As with previous comments, it is important to consider aspects outside the model. This model focuses on the oxygenation related aspects of vasculature, and although these are an important function of vasculature, they are not the only ones. Blood vessels are also important for delivering drugs, including bevacizumab and the buffer therapies considered earlier, and allowing greater immune system access to the tumour. However, they can also be co-opted to give the tumour other growth factors and serve as a point of invasion into the rest of the body. Here I consider neither of these aspects of angiogenesis inhibition therapy, nor countless other considerations that need to be added to the spherical cow sketched in this post.

Archetti, M. (2013). Evolutionary game theory of growth factor production: implications for tumour heterogeneity and resistance to therapies. British Journal of Cancer, 109(4): 1056-1062.

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

Archetti, M., Ferraro, D.A., & Christofori, G. (2015). Heterogeneity for IGF-II production maintained by public goods dynamics in neuroendocrine pancreatic cancer. Proceedings of the National Academy of Sciences of the United States of America, 112(6): 1833-8.

Bennett, C. L., Silver, S. M., Djulbegovic, B., Samaras, A. T., Blau, C. A., Gleason, K. J., … & Edwards, B. J. (2008). Venous thromboembolism and mortality associated with recombinant erythropoietin and darbepoetin administration for the treatment of cancer-associated anemia. The Journal of the American Medical Association, 299(8): 914-924.

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

Gatenby, R.A., & Gawlinski, E.T. (2003). The glycolytic phenotype in carcinogenesis and tumor invasion: insights through mathematical models. Cancer Research, 63 (14), 3847-54 PMID: 12873971

Gatenby, R. A., & Gillies, R. J. (2004). Why do cancers have high aerobic glycolysis? Nature Reviews Cancer, 4(11): 891-899.

Hemker, H. C., & Hemker, P. W. (1969). General kinetics of enzyme cascades. Proceedings of the Royal Society of London B: Biological Sciences, 173(1032), 411-420.

Mohyeldin, A., Lu, H., Dalgard, C., Lai, S. Y., Cohen, N., Acs, G., & Verma, A. (2005). Erythropoietin signaling promotes invasiveness of human head and neck squamous cell carcinoma. Neoplasia, 7(5): 537-543.

Raghunand, N., He, X., Van Sluis, R., Mahoney, B., Baggett, B., Taylor, C. W., … & Gillies, R. J. (1999). Enhancement of chemotherapy by manipulation of tumour pH. British Journal of Cancer, 80(7): 1005.

Ranieri, G., Patruno, R., Ruggieri, E., Montemurro, S., Valerio, P., & Ribatti, D. (2006). Vascular endothelial growth factor (VEGF) as a target of bevacizumab in cancer: from the biology to the clinic. Current Medicinal Chemistry, 13(16): 1845-1857.

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.

Warburg, O. (1956). On the origin of cancer cells. Science, 123(3191): 309-314.

Filed under Analytic, Commentary, Models, Preliminary, Technical Tagged with mathematical oncology, public good, replicator dynamics

About Artem Kaznatcheev
From the Department of Computer Science at Oxford University and Department of Translational Hematology & Oncology Research at Cleveland Clinic, I marvel at the world through algorithmic lenses. My mind is drawn to evolutionary dynamics, theoretical computer science, mathematical oncology, computational learning theory, and philosophy of science. Previously I was at the Department of Integrated Mathematical Oncology at Moffitt Cancer Center, and the School of Computer Science and Department of Psychology at McGill University. In a past life, I worried about quantum queries at the Institute for Quantum Computing and Department of Combinatorics & Optimization at University of Waterloo and as a visitor to the Centre for Quantum Technologies at National University of Singapore. Meander with me on Google+ and Twitter.