How cancer adds up

Mathematicians and surgeons are joining forces in the battle against cancer. Simon Hadlington describes how sophisticated modelling techniques can help clinicians to predict how a tumour could develop
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Michael Baum is professor of surgery at University College London and a leading cancer specialist. He has seen many developments in the field of oncology over a long career. But he confesses to being particularly excited about an approach to the disease which is being pioneered in Britain.

"I think this is absolutely revolutionary - it's going to be an important paradigm shift," he says.

Seth Schor, professor of oral cell biology at the University of Dundee, is equally enthusiastic. His particular interest is how cells migrate through tissue - a complicated process which is an important part of, for example, wound-healing and the growth and spread of cancers. "It is very difficult to get a handle on such a complex system. This approach will help give us new insights into these processes."

What approach? Unlike most advances in the study of intricate biological systems, it does not rely on high-powered computers or biotechnology. It is more fundamental and more subtle. It is mathematics.

In his capacity as president of the British Oncological Association, Professor Baum is organising a conference later this year in which the mathematics of complexity will feature prominently. "We have got to the point where mathematicians and clinicians are talking to each other," he says.

He believes that applying what he terms "new mathematics" to clinical oncology could eventually result in revolutionary new ways of approaching cancer as a disease, and to new, effective treatments.

Likewise, Professor Schor says: "It is only recently that mathematicians and biologists have begun to talk to each other. By mathematically deconstructing biological processes it is possible to identify the important variables and give new ideas and insights into biological mechanisms."

One of the people providing those insights is Mark Chaplain, one of a handful of mathematicians in the UK who are specialising in applying their knowledge and expertise to complex biological phenomena.

Dr Chaplain and a team of postgraduates based at the department of mathematics and computer science at the University of Dundee are developing mathematical models of the growth of solid tumours. Such tumours begin as a spherical nodule of cells that obtain nutrients by simple diffusion from the surroundings. This is termed the "avascular" growth phase. As the nodule grows and becomes too large to be fed by diffusion, it secretes a cocktail of chemicals that instruct nearby blood vessels to proliferate and migrate towards it, a process called angiogenesis. The vessels eventually penetrate the tumour and feed it, enabling it to continue to grow and invade the surrounding tissue - the "vascular" growth phase.

Chaplain's team has derived a set of equations that describe how the avascular phase depends on various factors that influence the growth rate, such as the supply of nutrients or the presence or absence of substances that may inhibit growth.

The team has also modelled how, during angiogenesis, blood vessels form a dense network around the developing tumour. They incorporate variables such as the rate of proliferation of the "endothelial" cells that form the new capillaries, and the concentration of chemical messengers - or "growth factors" - released by the parent tumour.

"We are confident that we have got a very good mathematical model which is realistic in terms of what actually happens in the body," says Chaplain. By manipulating this model, the mathematicians can predict what effect a change in a given variable - a growth factor, say, - might have on the development of the tumour.

For Professor Baum, this ability to model mathematically all the subtle complexities of a growing cancer could radically alter some of the foundations upon which treatment is based. When a malignant growth is discovered in a patient, physicians must take account of the possibility that the cancer may spread to form new growths - metastases - in other parts of the body. This usually requires hormone therapy or chemotherapy.

Around 20 years ago, a model was devised to give physicians a basis on which to form their chemotherapy regimens. This model assumed that at the time of the diagnosis of the cancer the patient's body contained "micrometastases" - hypothetical, invisible units of potential new cancer growth which could be destroyed by a period of chemotherapy. A mathematical relationship was calculated between the rate of growth of the micrometastases, their "killing capacity" and the effect of chemotherapy. The relationship was broadly linear: increasing the dose of chemotherapy would kill proportionally more malignant cells.

"But despite the fact that the model seemed very compelling, over the past 20 years we have made only very, very modest progress with survival rates," says Professor Baum. "One school of thought says that there is nothing wrong with the model - that the chemotherapy simply needs to be increased. An alternative view is that this model is wrong. When you look at what actually happens to patients, it just does not fit the maths."

That does suggest an important gap in the clinical treatment of cancer: that people are receiving incorrect treatment for their illness. There is no suggestion of negligence on the part of the doctors, who are carrying out their work in good faith. It is just that the maths hasn't kept pace. Professor Baum believes that the existing model is simplistic, and does not sufficiently take into account the enormous complexity of the system.

"We have tried to develop an alternative model, considering these putative micrometastases as complex organisms in a state of dynamic equilibrium. They contain cancer cells, vascular cells, host cells, and so on. We are now talking about the mathematics of complexity. What I am hoping to do, in collaboration with mathematicians, is factor in all the known variables that may determine whether micrometastases are successful or unsuccessful, so that the model better describes what we observe."

It is a tall order. But it is already becoming apparent that the new, complex models can predict the course of growth of a cancer far more realistically than can the existing linear models. The benefits of tinkering with equations should show up in the most useful way: more people undergoing successful cancer treatments. Some day, mathematics could save your life, too.

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