Oncology, the study of cancer, is just one of many specialisms which increasingly employs the predictive power of applied mathematics. This issue we chat with Trachette Jackson, professor of mathematics at the University of Michigan, to learn about the surprising effectiveness of mathematics in the treatment of cancer, as well as to hear about her own journey into mathematical oncology.

## Modelling in medicine

Trachette starts by bringing us up to date on how mathematics has been used in cancer treatment. “The mathematical approach has been applied to just about every aspect of tumour growth, starting decades ago.” One aspect is cancer therapeutics: “We write down equations that describe the mechanism of action of new drugs and how the tumour responds, then make predictions about how best to deliver those drugs.” Another aspect is more fundamentally biological, such as how cells are transformed: “You can find mathematical equations about the probabilities of acquiring mutations and under what circumstances a tumour forms, as well as what the composition of that tumour will be and how many cells in that tumour will have these different mutations.” This sort of modelling allows us to diagnose or assess the risk of cancer developing, as well as treat it.

Trachette starts by bringing us up to date on how mathematics has been used in cancer treatment. “Mathematical approaches have been applied to just about every aspect of tumour growth, starting decades ago.” One aspect is cancer therapeutics: “We write down equations that describe the mechanism of action of new drugs and how the tumour responds, then make predictions about how best to deliver those drugs.” Another aspect is more fundamentally biological, such as how cells are transformed: “You can find mathematical equations describing the probabilities of acquiring mutations and under what circumstances a tumour forms, as well as what the composition of that tumour will be and how many cells in that tumour will have these different mutations.” This sort of modelling allows us to diagnose or assess the risk of cancer developing, as well as treat it.

Trachette’s recently published paper, on the modelling of a drug targeting a chemical needed by cancerous tumours to grow, demonstrates the value of mathematical modelling. “We built a mathematical model that could look at the cross-talk between tumour cells and endothelial cells that secrete the chemical, then used it to predict responses of the tumour to this new treatment, along with traditional chemotherapy.” The model predicted something unexpected—administering the two drugs at the same time, as suggested by experimentalists, “would give you the worst results possible. Instead, the model predicted that fortnightly injections of chemotherapy along with continuous weekly administration of low amounts of targeted drugs would lead to synergistic responses.”

Could these same achievements have been made without the models? “I think the answer is probably yes. But it would be very expensive, and it would take a lot longer. What does math do? It speeds it up, and it saves the lives of millions of mice these drugs are tested on.” We were happy to hear, then, that mathematics is fundamentally better than biology and will soon be replacing it entirely. But Trachette kept us grounded by reminding us that it goes both ways. In fact, it is through her collaboration with experimentalists that she decides where to focus her attention. “They’re the ones who know what the big questions in cancer biology are, and which problems they would love answers to; those they think mathematical modelling might help them address.”

We couldn’t hold in the burning question any longer—what kind of maths are we talking about here? “I mainly use differential equations, both ordinary and partial, and sometimes delay differential equations.” In a delay differential equation, the derivative at a certain time is given in terms of the value of the independent variable at previous times. “I also use hybrid models, which are methods that combine continuous differential equations with a discrete approach, to answer questions that look at the interplay between something that diffuses like a chemical, the biochemistry, with the biomechanics, like a cell actually moving on tissue structures.”

Instead of having a blanket regime for every type of cancer, we really start looking at what to do for individual patients.

So, it all boils down to solving differential equations. Should be easy enough? “I’m talking about systems of non-linear differential equations, so we’re past the point of pen and paper analysis. We have to go to the computer and run simulations.” But that doesn’t stop Trachette from having fun. “Sometimes, we can do some model reduction to get the system in a form where we can say something about it analytically, but often the systems I work with require solving numerically.”

In the future, she expects to see big improvements in how patients’ data is used in mathematical modelling, in what is called ‘personalised medicine’. “When a patient is diagnosed with cancer, they’re put through a battery of tests that gives information about the molecular details of that cancer. We need much more of that data collected, and we need to be able to use all the information clinically available: from a specific person’s genetics, to the way that their body processes these drugs. Instead of having a blanket regime for every type of cancer, we really start looking at what to do for individual patients.”

## A leopard can change its spots

Trachette credits her parents for fostering her love of learning. “They really stressed the importance of education. They let me know that learning is fun, it’s OK to be a smart girl, and it’s OK to be a girl who is good at math.” She moved around a lot as a child, because her father was in the US air force, but she eventually settled in Arizona where she went to high school and became interested in mathematics. She went to a local university to take classes in the summer, and this is where she met her first maths professor. “He didn’t look anything like I thought a math professor would look. I was shocked that a normal looking person could be a mathematician. He became a very big role model.”

They let me know that learning is fun, it’s OK to be a smart girl, and it’s OK to be a girl who is good at math.

This professor helped to point her in the right direction when she went on to study for her degree at this same university. Initially, she wasn’t studying maths. “Believe it or not, I was an engineering major. I quickly learned I didn’t like engineering, so he called me into his office and said you should be a math major.” She took the advice, but at first chose pure maths, studying “nothing applied at all”.

It took a second role model to show her what applied mathematics can do and to inspire her to go into mathematical biology. James Murray, a pioneer in the field of mathematical biology, came to her university and gave a talk about how leopards get their spots. “I thought, ‘This does not belong in the math department! How is math going to have anything to do with leopards and spots?’ But I went to the talk, and…my mind was blown.”

James Murray later became her PhD advisor. At the time, she didn’t understand all the mathematics in his talk, but she took away what turned out to be a much more important message. “I understood that math can play a role in science, in biology, in medical science, and developmental biology, and I had never seen that before. That was when I knew I wanted to be an applied mathematician, because I saw real benefit of math in those areas. I wanted to be like this person I saw giving this talk.”

## Dreaming of the future

Our conversation turns onto the topic of diversity in academia. Trachette hopes that today’s young women can be inspired the way she was to consider mathematics. She says she is a “big fan” of recent initiatives to get more women interested in mathematics and has a lot of praise for the Association of Women in Mathematics in the US and their networking schemes, saying they are “just amazing”. She says they have communities where “women can find like-minded women in their own areas and form networks”. She is also very impressed by their mentoring programmes, which enable women to “find women at higher career stages, so they can ask those difficult questions through all the critical transitions in your career. Sometimes it feels a little isolating to be a woman in math. And this is a way to get, even at undergraduate level, connected to women.”

There needs to be proponents of diversity in each department to try to locally start changing the culture to be more welcoming and inclusive.

Trachette is very passionate about increasing opportunities in mathematics for people of colour. “Being an African-American mathematician at a research institution is very rare, and I want to change that demographic.” This is why she “started a programme to help underrepresented minorities progress through the pipeline. I think it’s very important to support the movement of diverse people through the PhD and beyond, so that we actually change the face of the professoriate.”

Trachette takes lots of opportunities to be a visible role model in her career, for example by appearing in *Power in Numbers* by Talitha Williams, which tells the stories of over 30 women’s journeys through maths. You can find Chalkdust’s review here.

She sees two approaches to improving diversity in universities. First, there is the “top-down” approach, involving national level organisations. “I know these big societies have diversity, equity, and inclusion committees, and are aware of the problem. The mathematical community often looks to these societies, so it trickles down.” But change can also be from the bottom-up, and it is important that more effort is made to make these kinds of improvements. “There needs to be proponents of diversity in each department to try to locally start changing the culture to be more welcoming and inclusive, and to take seriously the issue of recruiting a diverse graduate student body and faculty. I think the mathematical community really needs a deep dive into where we are and how we got here and what we can do change it.”

She seems positive about the future, and her closing message about the next generation of female mathematicians is very inspiring. “I hope they will blossom and find networks of other people to work with and find mentors at all levels. I hope that they find people to give them good advice, and to advocate for them, so they can eventually be agents of change.”