Game theory increases the chances of surviving metastatic cancer

Not all cancer cells are the same. They can work together, but also against each other – which is one of the reasons why the principles of game theory can be used in cancer therapy, says mathematician Kateřina Staňková. To put these findings into practice, Staňková – who is a graduate of VSB – Technical University of Ostrava (VSB-TUO) and a researcher at Maastricht University, the Netherlands – works together with the Moffitt Cancer Center in Florida, which has the only department of integrated mathematical oncology in the world.

Her original intention was to study theoretical mathematics in Prague, but as her parents wanted her to stay home in Ostrava, she enrolled in applied mathematics at the local technical university –a decision she is now very grateful for: “The favourite part of my studies was towards the end, when I started working with Professor Zdeněk Dostál at the Department of Applied Mathematics and took my first steps in research in the field,” says Kateřina Staňková.

Both your research and your courses focus on game theory. Could you explain what game theory is?
Classical game theory analyses situations where individuals interact with each other. While each has their own goal and the ways to achieve it, these goals also depend on what the others decide to do. All this can be described in mathematical terms and then you can start working on the best course of action for each of the parties in the given situation. A standard example of classical game theory is the prisoner’s dilemma. Imagine a situation where two prisoners, each in a separate room, are given the following choices: “If you confess while the other doesn’t, he will get thirteen years and you walk free. If you both confess, you’ll each get nine years. And if neither of you confesses, you’ll each get two years each.” If you paint this picture to them the right way, you can essentially make them confess. Since neither of them knows what the other is going to do, they automatically assume that the other will only do what is the best for themselves, leading to confessions from both. Obviously, it is not always this simple.

There is also dynamic game theory, which describes situations that evolve in time. Me, you, and everybody else can change their mind and the situation we are in also changes... At the same time, my goal – and other people’s goals – can depend on the decisions made by the others and on the specifics of the various situations. Dynamic game theory analyses what the best strategies for all the involved parties are, their properties, and what it all means for the analysed system.

And how is this relevant to cancer?
One of the subfields of dynamic game theory is evolutionary game theory, which describes the behaviour of living organisms in mathematical terms. Essentially, it is a mathematical representation of Darwin’s theory of natural selection. Not all cancer cells are the same: there are various types that can fight against each other, but also cooperate at the same time. For example, they might be competing for sugar and other nutrients found in the blood and working together to overcome the immune system. Evolutionary game theory describes how organisms react to what we do to them, or to what other organisms are doing, and how they adapt to proliferate and survive.

And who is the winner?
In the case of small organisms and cells, the winner is always whoever reproduces the quickest, whoever has the most offspring. The point is that if they can obtain a property that helps them reproduce quicker, this property will start spreading in the population. If, on the other hand, the mother cell has a property with the opposite effect, then this property will be eliminated from the population. A good example of this is metastatic cancer, or more specifically, prostate cancer. Let’s say that a patient suffering from prostate cancer has already had his prostate gland removed, but the cancer has spread or metastasised to the bones. At this stage, he probably only has several months left to live. Classical therapy would try to remove all testosterone from his blood. This is because all cells in the prostate gland, whether healthy cells or cancerous cells, have adapted to life in the prostate gland and need testosterone to live. This led doctors to believe that removing all the testosterone would kill all the cells that need it, including the metastatic cancer cells in the bones.

I’m guessing that unfortunately, it doesn’t work...
Right. Cancer cells react to this treatment very rapidly and mutate into cells that produce testosterone. So, you would need another type of therapy, this time killing the cells that produce the testosterone. Moreover, the treatment is highly toxic and often must be abandoned due to the side effects. And unfortunately, it enables cancer cells to mutate into cell types that do not need testosterone, essentially “teaching” them to live without it. We can offer a mathematical explanation of why it works like this.

How would game theory describe the situation you just mentioned?
You choose one drug, such as Lupron (chemical castration), with the idea that you are going to kill all the cells, and you apply the drug in the maximum tolerated dose, which essentially means a dose that won’t kill the patient. It is quite common, and this is something doctors don’t often talk about, that patients die not because of their cancer, but as a result of the toxicity of the drugs used to treat it. The game theory model shows that unless I kill all of the cells, I am training the biggest enemy I could have. According to the protocol used in metastatic cancer, doctors apply the maximum tolerated dose of the same drug over and over, until there is an inevitable evidence of either too high toxicity or tumour progression. However, cancer cells keep adapting to what we do to them and become more and more resistant. In this way, we are training a system that we cannot control, because after killing all the cells that react to the treatment, we are left only with those that do not react. We cannot use the same drug again, so we use another one, again at the maximum tolerated dose. After a while, this cannot be used anymore either, because the cells are also resistant to it. Eventually, you reach the point where you are out of options and you are fighting against cancer whose cells are resistant to anything you might use against it. Using mathematics, we are now trying to convince oncologists, radiologists, and other doctors that this is not the right way to go.

Have you had any feedback so far?
We recently had a paper published in JAMA Oncology, which is a traditional journal for oncologists and clinical physicians. We were quite surprised that they started listening to us.

You are working on your research together with the Moffitt Cancer Center in Florida. Why them?
This is the only institution in the world that has a department of integrated mathematical oncology. They listen to mathematicians and try to put our research into practice through clinical trials. It was the first place in the world where they did a prostate cancer clinical trial based on game theory.

Can you describe how these clinical trials work?
Every patient included in a clinical trial has their PSA levels measured – that’s an antigen that shows how widely the cancer has spread in their body. The standard aggressive treatment designed to kill the testosterone-producing cells is only applied until the PSA drops to half of the initial level. Afterwards, the therapy is stopped and you wait until the cancer “grows” all the way back and then start the treatment again. The overwhelming majority of patients who undergo this treatment are still alive after three years, which means that they have already lived three times longer than expected. This is because resistance against the treatment takes its toll on the cells: the cells that develop drug resistance are reproducing slower in the absence of the drug than non-resistant cells. The system fights against itself: when you apply the treatment, the resistant cells start winning; when you don’t, the non-resistant ones take over.

Did you set yourself a goal when you started the trials?
We knew that we could never actually defeat cancer using the standard therapy, that sooner or later it would stop responding to treatment. However, we wanted to find a way to turn this disease into a chronic condition, similar to what happened with HIV. We also had to work out how to surprise the cancer. The tumours are different in every patient, so the therapy is also different: some receive it every three months, others every six months or every twelve months.

What are your plans for the future?
We are currently opening clinical trials that will test using several drugs at the same time. Another area we want to focus on is helping patients with thyroid cancer. Non-metastatic thyroid cancer is easily operable and only a small percentage of patients progress into the metastatic phase. However, once that happens, they usually die within a year. We want to try to change this prognosis. We are also planning to start using chemotherapy as part of evolutionary therapy. We feel it is probably not going to be ideal to start the chemotherapy treatment only as a last resort at the very end. And once we are 100% certain how cancer works and how to control it, we can move on to the next level. Hopefully, we might be able to kill it completely in the next step, but we are not there yet. Therefore, there is no point in trying to kill all the cancer cells when we know this never works in patients with metastatic cancer.

Are any other departments working on the use of game theory in cancer therapy?
Besides the Moffit centre, I know that this topic is researched at the Johns Hopkins Medicine teaching hospital, by experts at the Mayo Clinic, and at the US hospitals that are part of the ORIEN network. In general, most of the research is done in the US and is only slowly spreading into Europe, where we are still at the stage of cell culture experiments outside the human body.

Have you ever visited the centre in Florida?
Yes, thanks to an EU grant, which is designed to facilitate cooperation between European institutions and institutions in the US and Canada. Since 2016, my PhD students and I have been able to visit Moffit regularly. The research centre is right next to the hospital, so you actually meet the patients. Although it is depressing, we can clearly see the purpose of our work. It is wonderful to know that we can help those patients.

Last November, I also visited Moffit to participate in a competition organised by the Integrated Mathematical Oncology department. At the beginning, they build teams composed of mathematicians, oncologists, radiologists, and other experts. The teams are given the topics on Sunday and then have five days to build a model for solving the problem. On Friday, the organisers evaluate the solutions and select the winning team. This year was the first time I participated and co-led one of the teams – and we won! We received $50,000 to continue developing our model for treating metastatic thyroid cancer.

You graduated from applied mathematics at VSB – Technical University of Ostrava. What were your studies like?
I must admit that in the beginning, I was less than the ideal student. (laughter) I always used to have straight A’s at secondary school without even trying too hard. Once I enrolled at the university, I didn’t really know how to study to pass an exam. At one of my first maths exams, I think it was functional analysis, I actually got an E (and was really happy to have passed at all)! However, the professor who led that class, Jiří Bouchala, really helped me back then: he essentially held a mirror up to me and helped me realise that I wasn’t as good as I thought. Thanks to him, I started digging deeper, trying to understand the basic principles in the field. So, I would like to thank him and apologise for how bad a student I was at the beginning. But I hope I have improved a little over time. (laughter)

Why did you decide to complete your PhD in Delft?
Originally, I came for three months as an Erasmus+ exchange student when I was writing my master’s thesis. My topic was convex optimisation and, in the Netherlands, they were using different algorithms to the ones I knew from Professor Dostál. So, I marched into the office of a renowned professor like a confident know-it-all and started explaining, in my appalling English, that we can simply do it better in Ostrava. To my surprise, I did not unnerve him; instead, he started a debate with me. He recommended a master-level course that he was teaching so that I could learn what they were doing. However, the course was in Dutch... The only thing that was in English was the presentation slides. But I think I was the only one who got ten points out of ten from the exam because I really wanted to understand the topic.

When I completed the course, he suggested that I should go and see another person at the university, a game theory researcher, about this topic. He actually recommended me to him as a potential PhD student, even though I knew nothing about game theory back then! (laughter) We chatted about what he was doing, what I was doing, and about game theory. At the end of the meeting, he offered me a PhD student position. And since I fell in love with Delft just as much as with game theory, I agreed.

What was it like to be a PhD student in the Netherlands?
It was an intense experience. Right at the start, my supervisor told me that he would be going away for two months and that there was nobody to supervise his master’s degree students so I would have to stand in for him. I was really worried that if I didn’t understand the topic well enough, I wouldn’t be able to do it. It was a great move on his part because it made me learn everything really fast. While I was studying for my PhD, I met my partner, who is now a professor in Delft, and so I extended my stay in the Netherlands a bit longer than I originally planned…

What made you decide to focus on cancer therapy?
My PhD thesis focused on Stackelberg games. Heinrich Freiherr von Stackelberg was a German mathematician and economist, who studied situations where one of the players can influence the other players much more than the other way around. I then applied Stackelberg games to the Dutch toll system. When you know how drivers make decisions and which routes they choose, you also know how to adjust the tollbooths on the roads to raise the most money or to relieve congestion.

I gradually moved on from this topic to biology and started focusing on the resistance mechanisms in pests. For example, leaf-eating pests are a significant problem with apple trees. And what did the gardeners do? They flooded them with all the chemical substances at their disposal in the highest possible dosage. As a result, a certain species was declared resistant to everything that we have. I worked on this with Professor Maurice Sabelis, who became a very close friend of mine, but died of cancer after three years. It was hard for me to deal with it. But since I was focusing on evolutionary game theory, I thought it might be a good idea to apply it to cancer.

You must have started learning a lot about biology...
You have to work as a part of a multidisciplinary team. For example, the paper on evolutionary therapy was co-authored by a biologist, a radiologist, and an oncologist who was responsible for the clinical trial itself. Obviously, I have to understand the basic principles, but to a certain extent, I will always be like a small child compared to the experts who study cancer every day.

Since you also graduated from a Czech university, what differences can you see between the Czech and the Dutch education systems?
University education in the Czech Republic is still very much focused on the individual student, to make sure that they learn as much as they can. And the stakes are also higher: if you fail a class for the third time, you will be kicked out of school. Maastricht University focuses on the application of knowledge rather than memorising information. Students work together on projects where they need to use their knowledge. VSB-TUO now also uses this system of teaching, but it is not as widespread. I don’t want my students to remember all the formulas; I can provide those. But I want them to be able to apply the formulas correctly.

You and your partner have two young children. How do you reconcile taking care of your children with a time-consuming job?
In the Netherlands, both parents usually participate in child-rearing. Each parent has at least one mama dag and one papa dag, which are days when the children are with their mother and days when they are with their father. It is quite demanding, but we are managing.