A new review explains how turning to mathematics is helping researchers decode one of cancer’s most elusive traits: its ability to evolve and adapt.

Understanding how cancers change over time is crucial for the development of effective treatment strategies. If scientists can identify exactly how cancer changes in response to different stimuli, they will be much better placed to tackle treatment resistance and disease progression. In the longer term, this could improve outcomes for patients with all types of cancer.

With the aim of expanding the use of mathematical modelling in this field, the authors have highlighted recent successful projects and considered further ways in which this approach could benefit teams.

The review was written by researchers in the Centre for Evolution and Cancer at The Institute of Cancer Research, London, and it was published in the journal Current Opinion in Cell Biology. The authors are supported by funding from Cancer Research UK and the Medical Research Council.

Understanding the dynamics of cancer evolution

Cancer evolution refers to the processes by which cancer cells change over time, allowing them to survive, grow and resist treatment.

This often occurs through genetic mutations that provide cells with a structural or functional advantage that they pass on to daughter cells. In time, this ‘stronger’ population can represent a significant proportion of the tumour, making it more difficult to treat.

However, non-genetic mechanisms can also shape cancer cell behaviours. In fact, cancer cells sometimes alter their phenotype – their observable morphological and physiological characteristics – in response to a change in environment, without any genetic changes occurring. This phenomenon, known as phenotypic plasticity, has been shown to play an important role in treatment resistance and cancer progression across different cancer types.

At the Centre for Evolution and Cancer, researchers are investigating the combined role of genetic and non-genetic mechanisms in shaping cancer cell behaviour.

A complementary tool

Mathematical modelling is used in various fields, including economics and social and natural sciences. This tool typically presents processes or scenarios in the form of equations, which can provide new insights into complex mechanisms and relationships. It can also serve as a platform for experts to test hypotheses, determine what is likely to happen if certain parameters change and design future experiments.

The review authors believe that mathematical modelling should be used more widely in cancer research to complement traditional experimental approaches. In their work, it facilitates the reconstruction of evolutionary changes over extended periods.

First author Chloé Colson, Postdoctoral Training Fellow at The Institute of Cancer Research (ICR), explained:

“Recent advances in single cell sequencing technologies have enabled us to study cancer cells at higher resolution than ever before. However, most available clinical data only reveal the traits of a cell at a single point in time.

“As cancer evolution is an inherently time-dependent process, we use mathematical models to help us create a more detailed picture of the changes – both genetic and non-genetic – that cancer cells undergo over time, including in response to treatment.”

A choice of models

The review provides a concise yet comprehensive overview of five diverse mathematical approaches that have been, and can be, used to learn about cancer evolution.

Each method has different strengths and limitations. For instance, stochastic branching processes (SBPs) – which simulate the life cycle of individual cells – can help uncover rare mutation events, but they require very detailed data. Meanwhile, ordinary differential equations (ODEs) – which model average population dynamics over time – are effective for predicting tumour growth under treatment, but they do not take individual variability into account.

By selecting the right combination of existing models, researchers can create a powerful toolkit for reconstructing tumour evolution. It is also important that teams continue to develop new, more advanced models to help them address their specific outstanding areas of interest.

Simulating how tumours evolve and respond to treatment will give researchers the knowledge they need to be able to design smarter strategies that anticipate resistance.

One approach currently under investigation is adaptive therapy – a treatment strategy that involves dynamically adjusting drug dosage based on how the tumour responds over time. Rather than eliminating the cancer cells, which often leads to drug-resistant cells dominating, the aim is to maintain a population of drug-sensitive cells that limit the growth of resistant ones.

Other techniques that can be optimised by a full understanding of cancer evolution include combination treatments that target multiple phenotypes at the same time and timed interventions that are delivered when they are most likely to disrupt cancer growth.

“These models aren’t just academic exercises”

For oncologists and researchers, this fusion of maths and medicine could be the key to staying one step ahead of cancer.

Chloé Colson said:

“We’ve put together a comprehensive review focusing on the role of mathematical modelling for studying both genetic and non-genetic modes of cancer cell evolution. We think the review demonstrates the value of mathematical approaches for understanding complex cancer biology.

“It’s exciting to think that as data from single cell sequencing and spatial imaging become more accessible, these models will become even more powerful. The fusion of biology and mathematics is poised to transform how we understand – and ultimately outsmart – cancer.”

Professor Trevor Graham, Professor of Genomics and Evolution and Director of the Centre for Evolution and Cancer at the ICR, and senior author of the review, said:

“These models aren’t just academic exercises; they’re shaping the future of cancer therapy. As we dive deeper into complex biological systems, mathematical modelling is emerging as a vital ally. The models give us a window into what happened in the past and enable us to forecast what the future holds for a disease that refuses to stand still.

“Ultimately, we aim to clarify the mechanisms that allow cancer to develop drug resistance and an increased ability to spread, so we can create more effective and evolution-aware treatment strategies.”

Image credit: Gerd Altmann from Pixabay