I am a translator. I translate from biology into mathematics and vice versa. I write mathematical models which, in my case, are systems of differential equations, to describe biological mechanisms, such as cell growth. Essentially, it works like this. First, I identify the key elements that I believe may be driving behavior over time of a particular mechanism. Then, I formulate assumptions about how these elements interact with each other and with their environment. It may look something like this. Then, I translate these assumptions into equations, which may look something like this. Finally, I analyze my equations and translate the results back into the language of biology.
A key aspect of mathematical modeling is that we, as modelers, do not think about what things are; we think about what they do. We think about relationships between individuals, whether they be cells, animals or people, and how they interact with each other and with their environment. Let me give you an example. What do foxes and immune cells have in common? They're both predators, except foxes feed on rabbits, and immune cells feed on invaders, such as cancer cells. But from a mathematical point of view, a qualitatively same system of predator-prey type equations will describe interactions between foxes and rabbits and cancer and immune cells.
Predator-prey type systems have been studied extensively in scientific literature, describing interactions of two populations, where survival of one depends on consuming the other. And these same equations provide a framework for understanding cancer-immune interactions, where cancer is the prey, and the immune system is the predator. And the prey employs all sorts of tricks to prevent the predator from killing it, ranging from camouflaging itself to stealing the predator's food. This can have some very interesting implications. For example, despite enormous successes in the field of immunotherapy, there still remains somewhat limited efficacy when it comes solid tumors. But if you think about it ecologically, both cancer and immune cells—the prey and the predator—require nutrients such as glucose to survive. If cancer cells outcompete the immune cells for shared nutrients in the tumor microenvironment, then the immune cells will physically not be able to do their job.
This predator-prey-shared resource type model is something I've worked on in my own research. And it was recently shown experimentally that restoring the metabolic balance in the tumor microenvironment—that is, making sure immune cells get their food—can give them, the predators, back their edge in fighting cancer, the prey. This means that if you abstract a bit, you can think about cancer itself as an ecosystem, where heterogeneous populations of cells compete and cooperate for space and nutrients, interact with predators—the immune system—migrate—metastases—all within the ecosystem of the human body. And what do we know about most ecosystems from conservation biology? That one of the best ways to extinguish species is not to target them directly but to target their environment.
And so, once we have identified the key components of the tumor environment, we can propose hypotheses and simulate scenarios and therapeutic interventions all in a completely safe and affordable way and target different components of the microenvironment in such a way as to kill the cancer without harming the host, such as me or you.
And so while the immediate goal of my research is to advance research and innovation and to reduce its cost, the real intent, of course, is to save lives. And that's what I try to do through mathematical modeling applied to biology, and in particular, to the development of drugs. It's a field that until relatively recently has remained somewhat marginal, but it has matured. And there are now very well-developed mathematical methods, a lot of preprogrammed tools, including free ones, and an ever-increasing amount of computational power available to us.
The power and beauty of mathematical modeling lies in the fact that it makes you formalize, in a very rigorous way, what we think we know. We make assumptions, translate them into equations, run simulations, all to answer the question: In a world where my assumptions are true, what do I expect to see? It's a pretty simple conceptual framework. It's all about asking the right questions. But it can unleash numerous opportunities for testing biological hypotheses. If our predictions match our observations, great—we got it right, so we can make further predictions by changing this or that aspect of the model. If, however, our predictions do not match our observations, that means that some of our assumptions are wrong, and so our understanding of the key mechanisms of underlying biology is still incomplete.
Luckily, since this is a model, we control all the assumptions. So we can go through them, one by one, identifying which one or ones are causing the discrepancy. And then we can fill this newly identified gap in knowledge using both experimental and theoretical approaches. Of course, any ecosystem is extremely complex, and trying to describe all the moving parts is not only very difficult, but also not very informative. There's also the issue of timescales, because some processes take place on a scale of seconds, some minutes, some days, months and years. It may not always be possible to separate those out experimentally. And some things happen so quickly or so slowly that you may physically never be able to measure them. But as mathematicians, we have the power to zoom in on any subsystem in any timescale and simulate effects of interventions that take place in any timescale.
Of course, this isn't the work of a modeler alone. It has to happen in close collaboration with biologists. And it does demand some capacity of translation on both sides. But starting with a theoretical formulation of a problem can unleash numerous opportunities for testing hypotheses and simulating scenarios and therapeutic interventions, all in a completely safe way. It can identify gaps in knowledge and logical inconsistencies and can help guide us as to where we should keep looking and where there may be a dead end.
In other words: mathematical modeling can help us answer questions that directly affect people's health—that affect each person's health, actually—because mathematical modeling will be key to propelling personalized medicine.
And it all comes down to asking the right question and translating it to the right equation...and back.