# Microenvironmental effects in prostate cancer dynamics

I am currently visiting David Basanta and Jacob G. Scott at the Moffitt Cancer Center in Tampa, FL. I primarily study evolutionary game theory for the sake of evolutionary game theory. Sometimes I am curious about how it shapes our understanding of classic game theory, and how it broadly connects to other fields. Occasionally, I wander into pondering on economic or human questions, or at least sociobiology. However, the biggest application of EGT for most people is in biology, originally in the study of fisheries of sex-allocation and now even in medical fields like cancer research. Although I have briefly alluded to work on cancer before, it took the personal encouragement of David and Jacob for me to look at the field carefully. Naturally, I started with their paper. Half a year ago, I started drafting this post — my first on cancer — but it got lost in the perpetual “I’ll finish it up tomorrow” pile.

With only thirty minutes to spare, ‘tomorrow’ is finally today.

Over the last two days, David and I locked ourselves in his apartment and then office in the Integrated Mathematical Oncology Department. We discussed his papers on metastasis, the Warburg effect, and tumor-stroma interaction in prostate cancer — the topic of this entry.

Basanta et al. (2012) model the tumour-stroma interaction in prostate cancer through discrete-time simulation of inviscid replicator dynamics. They consider three types of cells: stroma (S) and tumor cells that are either microenvironmentally dependent (D) or independent (I). Due to biological considerations that I do not understand, this results in the payoff matrix for S, D, I:

$\begin{pmatrix} 0 & \alpha & 0 \\ (1 - \beta) + \alpha & (1 - \beta) - r\beta & (1 - \beta) + \rho \\ 1 - \gamma & 1 - \gamma & 1 - \gamma \end{pmatrix}$

Where $\alpha$ is the benefit from cooperating between an S and D cell, $\beta$ is the cost of extracting resources from the microenvironment, $\gamma$ is the cost of being microenvironmentally independent, and $\rho$ is the benefit derived by D from growth-factor produced by I cells. At this point, I must note that the authors restrict to $r = 1$ but don’t justify why two Ds interacting results in twice the extracting cost instead of the same or any other arbitrary factor $1 + r$. I assume it is something like depletion of the microenvironment, but I might be wrong.

For accessibility reasons, in the body of the paper the authors decide to use discrete-time simulation, but since this blog is not a journal for physicians, I will stick to doing replicator dynamics analytically. First, we will look at the dynamics on the edges of the simplex when one of the strategies is missing, i.e $p = 0$, $q = 0$, or $p + q = 1$. Look at the three submatrices:

$\begin{pmatrix} 0 & 0 \\ 1 - \gamma & 1 - \gamma \end{pmatrix}, \quad \begin{pmatrix} 0 & a \\ (1 - \beta) + \alpha & (1 - \beta) - r\beta \end{pmatrix}, \quad \begin{pmatrix} (1 - \beta) - r\beta & (1 - \beta) + \rho \\ 1 - \gamma & 1 - \gamma \end{pmatrix}$

We will assume that $0 \leq \alpha, \beta, \gamma, \rho \leq 1$ (as the authors do). Looking at the faces:

1. I-S: this face always flows towards all-I.
2. I-D: this face has three distinct dynamics:
1. if $\gamma \leq \beta - \rho$ then the flow is towards all-I.
2. if $\beta - \rho < \gamma < (1 + r)\beta$ then the flow is toward a stable face fixed-point with a proportion $p_{ID} = \frac{(1 + r)\beta - \gamma}{r\beta + \rho}$ of strategy I.
3. if $\gamma \geq (1 + r)\beta$ then the flow is toward all-D.
3. D-S: this face has three distinct dynamics, but only two are physically realizable:
1. if $\beta \leq \frac{1 - \alpha}{1 + r}$ then the flow is toward all-D.
2. if $\frac{1 - \alpha}{1 + r} < \beta$ then the flow is toward a stable face fixed-point with a proportion $p_{DS} = \frac{1 + \alpha - \beta}{2\alpha + r\beta}$ of strategy D.

As in the case of the Warburg effect, we can now draw the six possible simplexes for each of the conditions.

The possible simplexes for prostate cancer dynamics. The coloured region marks the basin of attraction for the stable states. In the top row (condition 3.1) and no internal fixed points; the whole simplex is attracted to either all-I, $p_{ID}$ or all-D depending on the column parameters. In the second row (condition 3.2) it is possible to have an internal fixed point for some parameters in columns 1 and 2 (when 2.1 and 2.2 hold). If there is an internal fixed point then the purple region is flows to the $p_{DS}$ DS-face fixed point, and the blue region flows to all-I. If there is no fixed point then the purple region is of size zero, and all flow is toward all-I. In the bottom right simplex (condition 2.3 and 3.2), all flow is toward the $p_{DS}$ DS-face fixed point.

To fully understand the dynamics, we need to study the internal fixed point. There is no fixed point if condition 2.3 or 3.1 holds. However, in conditions 3.2 with $\gamma < (1 + r)\beta$ it is possible to have an internal saddle-point give by:

$\begin{pmatrix} \frac{\frac{1 - \gamma}{\alpha}(\alpha + r\beta) + \beta - \rho - \alpha}{\rho - \alpha} & \frac{1 - \gamma}{\alpha} & \frac{\alpha + \gamma - 1}{\alpha} - \frac{\frac{1 - \gamma}{\alpha}(\alpha + r\beta) + \beta - \rho - \alpha}{\rho - \alpha} \end{pmatrix}$

Where the first component is the proportion of I, the second is D, and the third is S. When it exists (i.e. all components greater than 0), the saddle-point divides the game space into two basins of attraction. They are shown in blue and purple. If the initial proportion of agents is the purple region then it converges towards the $p_{DS}$ equilibrium with the I agents going extinct. However, if it starts in the blue region then the dynamics are toward either all I (in condition 2.1) or $p_{ID}$ with all the D agents going extinct.

I am no doctor, I don’t even play one on the internet, so all of my speculation on medicine should be taken with a grain of salt. However, from conversations with David, I gather that the worst achievable results is a polyclonal tumor of microenvironmentally dependent and independent cells — the $p_{ID}$ equilibrium. The mixture of different types would be difficult to target, since it would require control over both the microenvironment and direct management of cells. On the other hand, the easiest case to treat (apart from no tumor at all, just S, but that is not achievable) is a clonal population of just microenvironmentally dependent cells. This would give you just one opponent that can be targeted in two ways: either by attacking the tumor directly or its environment. The second best outcome, would be a polyclonal population of healthy stroma cells (S) and D cells — the $p_{DS}$ equilibrium. Although it would complicate treatment slightly because of the presence of healthy tissue, you would still only have to target one type of cells (the Ds) and have two approaches available.

As such, the most interesting simplex to study is the middle column of the second row when conditions 2.2 and 3.2 hold and the internal fixed point exists. In that case, we have the tension of bifurcation between two equilibriums: either towards the worst achievable outcome or the second best. We have to be aware of the current proportion of tumor and healthy cells, and can alter the dynamics by targeting any of the four game parameters. Suppose that a patient’s current cell balance is in the ID basin of attraction (blue). Oncologists would need to intervene to move the patient into the PD basis of attraction (purple). The standard approach would be to try to selectively remove part of the tumor and alter the balance of cells so that the new population is the the purple region. Another approach, would be to alter the game parameters by controlling the microenvironment to move the fixed-point around so that the previously blue region that the patient is in becomes purple. Once the microenvironmentally independent cells disappear, the doctors would have a simpler tumor to treat.

Of course, this is not the end for modeling prostate cancer. Specifically, prostate cancer — more so than many other cancers — has a very structured composition. As such, this is an important candidate for the Ohtsuki-Nowak transform/ Unfortunatelly, that treatment is a little bit more involved, and you are probably tired of reading all this math — I will save it for another time. As I say to myself: “I’ll finish it up tomorrow.”

Basanta D, Scott JG, Fishman MN, Ayala G, Hayward SW, & Anderson ARA (2012). Investigating prostate cancer tumour-stroma interactions: clinical and biological insights from an evolutionary game. British journal of cancer, 106 (1), 174-81 PMID: 22134510