# Healing wounds with differential equations

My research field is mathematical biology. I’m a mathematician working on healthcare challenges, including wound healing and infectious disease management. I find it deeply fulfilling to meet people who enjoy better health due in part to my mathematical problem-solving.

I’ve always enjoyed books, board games and puzzles. I became interested in maths in primary school and it was my favourite subject throughout secondary school. I had a high-school teacher who encouraged me to do extension reading and exercises beyond the classroom maths curriculum.

During my Bachelor of Mathematics, I realised that I wanted to extend my understanding beyond graduation. I met my eventual PhD supervisor – Emeritus Professor Sean McElwain – halfway through second year and hearing him talk about modelling in biology strongly influenced my choice of research field. My current mentors include Professor James McCaw and Professor Kate Smith-Miles.

My research is about pulling back the layers of a biological problem to find underlying mathematical forms. Biological systems are complex, so they can seem messy and unpredictable. Mathematical modelling of biological systems helps with the prediction and control of outcomes.

For example, mathematical modelling has helped to guide responses to the COVID-19 pandemic. We can model how an infectious disease is transferred from person-to-person and simulate different scenarios – like isolating patients for varying lengths of time – to predict the impact on disease spread.

Or we can model how cells communicate and affect each other during the healing process and investigate how different wound treatments can slow or accelerate healing – helping doctors to determine the best treatment and minimise unwanted side effects.

Skin ulcers can become a problem when changes in the veins lead to poor circulation and increased susceptibility to infection. I’ve collaborated with medical researchers at the Alfred Hospital to model how ulcer healing is affected by applying pressure bandages in various ways.

I’ve also applied mathematics to optimise hyperbaric oxygen therapy for diabetic wounds. Placing the patient inside a chamber containing pure oxygen at higher-than-normal air pressure helps their lungs to transfer more oxygen into the bloodstream, which helps the body fight infections and heal skin ulcers.

We used mathematical modelling to help determine the optimal pressure in the chamber, time spent inside and the frequency of repeated visits.

The University of Oxford has collected data on how drug resistance has spread through malaria parasites. I’m using this data to create mathematical models that can help to predict the spread of resistance while also collaborating with the Melbourne School of Population and Global Health on studies of malaria and other infectious diseases.

British mathematician Alan Turing – famous for his work on the Enigma machine – was a pioneer in my field. In the 1950s, he showed how mathematical equations govern animal pigmentation patterns. Turing patterns are also seen in embryo development and other genetic expressions.

Like Turing, I work to create mechanistic mathematical models of dynamic biological systems – like a wound under treatment – by applying ordinary and partial differential equations.

Differential equations can tell us a lot. We can use differential equations to quantify how a variable of interest (e.g. wound healing) changes as we vary input parameters (e.g. pressure bandages or oxygen treatment) over time.

Ordinary differential equations have a single independent variable (e.g. modelling how treatments affect healing over time) and partial differential equations have two or more independent variables (e.g. modelling how treatments affect healing of wounds of different depths over time).

Finding the right biologists to collaborate with can be challenging. They need to be willing to sit down and talk for a long time and be patient when you ask – from their perspective – silly questions. The potential return on this investment of their time is that mathematical modelling can deliver insights that they couldn’t achieve, or that would take much longer to achieve, through biological experiments alone.

My research funding comes from a variety of sources, including the Australian Research Council (ARC), the National Health and Medical Research Council, and the Bill and Melinda Gates Foundation. I’m also involved in a new ARC Centre of Excellence for the Mathematical Analysis of Cellular Systems, which has been funded for seven years.

Around one in four Australian PhD students in mathematics are women, but in mathematical biology at the University of Melbourne around half of the PhD students are women. In my opinion, many women are drawn to careers where they feel they can make a real difference to people’s lives and mathematical biology offers that. Not that men aren’t interested in making a difference, but women tend to dominate the ‘caring’ professions, like teaching and nursing.

We sadly still hear examples of active sexism, but awareness of gender bias and disadvantage is now front-of-mind for most recruitment, funding and award panels, and initiatives that address systemic inequity are increasing. For example, I was proud to receive a Women in Science Emerging Researcher Award from the Faculty of Science at the University of Melbourne.

I chair the Australian Mathematical Society’s Women in Mathematics Special Interest Group, which aims to support women, trans and gender-diverse people to achieve their potential in all areas of mathematics.

My advice to young people wondering whether a career in mathematics might suit them is don’t base your decision entirely on the secondary school curriculum. At high school, you’re learning vital foundations.

Maths gets much more interesting and powerful at university and you can specialise according to your interests.

Studying maths can take you down so many different avenues. Research, teaching, finance and insurance are obvious pathways, but every industry and government agency needs graduates with mathematical skills.

If you want to enhance life length and quality for lots of people, come and study mathematical biology with me.

– As told to Rebecca Colless

12 May is International Women in Mathematics Day. This date commemorates the birthday of Maryam Mirzakhani, a world leader in geometry and dynamical systems before her death at 40 from breast cancer.

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