# Mathematical Biology

Mathematics gives new hope to cancer researchers

Pancreatic cancer is one of the deadliest cancers in the world. If someone is diagnosed with it, there is only a 25% chance that they will survive the next year and a 6% chance that they will still be alive after five years. This is because the cancer is often at an advanced stage before it is discovered. But why is it not detected earlier?

For scientists, there are only two possible explanations. Either the cancer grows very quickly, or it grows slowly but shows no symptoms for a long time. Knowing which of these reasons is the correct one is extremely important for doctors and pharmaceutical companies, not to mention the patients. If the cancer is simply spreading very quickly then there is little hope of stopping it other than with the usual methods of chemotherapy or radiation therapy. However, if it is silently growing for a long time then hospitals could implement regular screening processes which could potentially catch the growth early and save thousands of lives.

Dr Tibor Antal is a mathematician who is currently studying exactly these sorts of questions. Together with a team of researchers at Johns Hopkins University and Harvard University, led by biologist Chris Iacobuzio-Donahue, he thinks he is starting to get some of the answers.

Tibor’s area of expertise is in stochastic processes; that is, processes which have a degree of randomness in their behaviour. His favourite example, and one which he studies, is that of mutations in DNA.

Our bodies are constantly regenerating themselves, with each cell dividing into two daughter cells about once per week. However, the process of copying a cell’s DNA into two new cells is not perfect, and on average there will be one error made for each division that happens. This copying error is called a mutation. The vast majority of mutations are completely harmless; some will stop the cell from working properly and so get killed off very quickly. In rare cases, the mutation leads to the cell becoming cancerous.

A person’s body is normally in a state of equilibrium: the number of cells dying is equal to the number of new cells being created. If something upsets this situation (for example, a freak mutation) so that there are more cells being created than destroyed, then the person develops a tumour. Tumours can grow for a long time without being dangerous to the person, but as soon as they start affecting the surrounding tissue then we call it a cancer.

Tibor’s research involves making mathematical models of these mutations and using the models to predict the growth of a tumour. If a cell gets a driver mutation that allows it to divide more rapidly, then the daughter cells will inherit this driver mutation and so they will also divide more rapidly. This means the process will become exponential, so that the tumour grows faster and faster as time progresses. As well as being able to predict how fast a tumour will grow, Tibor’s models also allow him to work backwards and to predict how old a tumour must be from knowing the number of driver mutations in it.

This brings us back to the question about pancreatic cancer and why it is so deadly. Tibor, along with two Harvard researchers Martin Nowak and Ivana Bozic, was able to take data about mutations in the cells of patients who had died of pancreatic cancer and to run them through his model to predict how old the cancer was. The results were astounding.

Contrary to anyone’s expectations, he calculated that pancreatic cancers are generally about ten years old before they are discovered! For years, these mutated cells grow inside the patient, causing no symptoms at all, before finally spreading into other organs and causing havoc. By this point, the cancer is so large that surgery is impossible.

This research has finally given us hope. If pancreatic tumours can grow benignly for ten years, then this gives doctors ten years in which to detect and remove them before they do any damage. This is not to say that the problem has been solved. Good, non-invasive, screening procedures for tumours of the pancreas don’t yet exist, but Tibor’s result gives us a reason to put more funding into figuring out this next step.

We can use the same mathematical models to investigate the growths of other types of cancer, and even to investigate different types of mutations. Tibor’s latest research is looking into the mutations in mitochondrial DNA. Mitochondria live inside cells and provide the cells with energy, so they are very important to our bodies, but at the same time they are even more prone to DNA mutations than the cells themselves. There are a wide range of diseases which arise as a result of these mutations, including diabetes, dementia and some forms of deafness, although it is unknown whether mitochondrial mutations could be linked to the growth of cancers. Hopefully the next few years of research at Edinburgh will help us answer some of these questions and continue the fight against cancer.

About our mathematician: When Tibor Antal is not working on new models for studying cancer, he is often found playing jazz on his guitar. His dream is to one day play in the rock band Wild Type, which consists completely of cancer biologists and was started by his collaborator Bert Vogelstein. Bert originally trained as a mathematician and is now one of the most highly cited scientists in the world.

Read other articles about Tibor's research by the BBC, the New York Times and Science Daily.