This chart could save your life: How mathematicians are fighting the war on cancer

Today, statistics don't just give cancer patients their chances of survival. As mathematicians get to work on our genetic data, they are opening up new fronts in the war on cancer.

Last year Andrew Jackson became another cancer statistic. At the age of 56 he was diagnosed with tumours in his prostate gland and given a 50-60 per cent chance of a cure by his consultant radiologist.

Andrew was one of about 35,000 men in Britain who in 2012 were told they had prostate cancer. What he really wanted to know was whether he would become one of the 11,000 men who die each year of the disease. He has a 40-50 per cent chance of falling into that statistical bin.

Cancer, perhaps more than any other disease, is a game of numbers. Anyone newly diagnosed wants to know their chances of surviving an illness that will affect one in every three people at some point in their lives. Some 11 million people in the world are diagnosed each year with cancer, and 7 million die of the disease.

The statistics predict that Andrew has a slightly better chance of surviving his "locally advanced" prostate cancer than winning the flip of a coin. "I'm an optimist and the way I see it, the glass is more than half full," he says.

Andrew (not his real name) is one of several thousand men taking part in research aimed at ending the lottery of cancer survival. His genome – the entire three billion chemical "letters" of his genetic code – is being scanned by the powerful tools of DNA science for a greater understanding and better treatment of his disease.

Over the past decade, since the decoding of the first human genome in 2003, a quiet revolution has taken place within cancer-research centres. It took more than a decade and about £2bn to decode the first genome. Now it takes a few hundred pounds and a couple of days.

Cancer is the quintessential disease of the genes because all tumours begin their career with changes to the DNA of one of about 50 trillion cells of the human body. Yet the genetics revolution has revealed a hidden complexity to cancer that few people had anticipated.

Genome research has shown that cancer is a chaotic and dynamic foe capable of dodging the supposed "silver bullets" of cancer science. Unravelling this chaos has produced vast quantities of data – big numbers that need to be sifted, analysed, and understood.

"I think what these studies show is that cancer is incredibly complicated, far more complicated than we'd like it to be," says Peter Campbell of the Wellcome Trust Sanger Institute in Cambridge. "We've always underestimated cancer. We had no window into its complexity. We were only looking at superficial features. Now with the genome we can go into far more detail," he says.

The devil, however, is in this detail. DNA sequencing has churned out unbelievable quantities of raw data, so much so that cancer researchers – steeped in the "soft" science of biology – are desperate for help. Mathematics, once described by Albert Einstein as "the poetry of logical ideas", has come to their rescue.

Genome sequencing has effectively taken cancer research from an era of still photography to videos, and mathematical modelling is deciphering the moving images. "We can get hundreds of billions of snapshots that give us a much finer picture of the cancer cell. It becomes an intensely mathematical and computational problem. Mathematics is going to pervade all aspects of cancer research, and possibly cancer therapeutics," Campbell says.

"I think that maths can give us a paradigm for testing a hypothesis. If we can build mathematical models of how cancer cells develop resistance to drugs, for instance, it may enable us to develop new treatments. Maths will be a component of the war on cancer," he says.

Number crunching has always, of course, played a role in cancer epidemiology. Even before President Richard Nixon declared war on cancer in 1971, scientists were using numbers to demonstrate links between cancer and its potential causes – tobacco and lung cancer being the most famous example. But never before has mathematics been so central to both fundamental cancer research and clinical treatment.

In April, for instance, Cary Oberije of Maastricht University in the Netherlands told the European Society for Radiology and Oncology that mathematical models have made better predictions about the outcome of cancer treatment on patients than the doctors who were actually treating them. "If models based on patient tumour and treatment characteristics already out-perform the doctors, then it is unethical to make treatment decisions based solely on the doctors' opinions," Oberije says. "We believe models should be implemented in clinical practice to guide decisions."

The same view is held by Robert Gatenby of the Moffitt Cancer Centre in Tampa, Florida. "Mathematics can help us to understand cancer and to treat it effectively we need to understand it at the most fundamental level," Gatenby says. "You might even one day see mathematicians in the clinic working with the clinician and the patients."

Gatenby has been working on constructing mathematical models of cancer since long before the genomic revolution. His thesis was that cancer is more than simply a lump of aberrantly expanding tissue. "Tumours don't just grow. They are more dynamic than that," Gatenby says. Neither is their intricacy to be feared. "Mathematical models show that the complexity of cancer is not some kind of magic or almost unworldly force," he says.

His early mathematical models of cancer attempted to emulate the reality of what happens when a rogue cell turns cancerous. He described in mathematical terms how cancer cells begin to compete for oxygen, how they produce lactic acid that poisons healthy cells, or how they stimulate the growth of blood vessels supplying nutrients to the growing tumour.

Gatenby used game theory of economics and the differential equations of evolutionary biology to predict how cancer cells compete with healthy cells. He used these models to calculate the doses and combination of drugs that will limit the chances of tumours developing resistance to f anti-cancer drugs. Instead of playing "whac-a-mole" with cancer by blasting it with one treatment after the next, he suggested a more strategic game of anticipatory chess.

Gatenby's models predicted ways of "outwitting" the blind evolution of cancer cells. By anticipating how tumours will react to drugs, Gatenby devised ways of outfoxing future mutations that impart drug resistance. "Cancer cells can only respond to the here and now. They cannot anticipate the future, but we can," he says.

Mathematical models of cancer can time shift to anticipate tumour development. Gatenby's colleagues at Moffitt have built a computer model of prostate cancer. They itemised some 250 different clinical markers of 650 prostate tumours – each sliced several times – at different stages of the disease. The result was a digital version of how a typical prostate tumour develops.

The computer model can be fast-forwarded in time, much like the "run" of a weather-forecasting model in a supercomputer. In doing so, the Moffitt scientists can produce predictions of whether a particular kind of tumour is likely to become aggressive – the risk of it spreading to other parts of the body.

Similar mathematical models are shedding important insights into pancreatic cancer, one of the deadliest of tumours. When someone is diagnosed with cancer of the pancreas it is almost always too late. A new patient has only a 25 per cent chance of being alive a year later, and just a 6 per cent chance of surviving after five years.

It used to be thought that this was because pancreatic cancer grows so rapidly. But computer models have in fact shown that pancreatic cancers develop slowly over many years, with most being at least 10 years old when they are first diagnosed.

These models were constructed by calculating when the genetic mutations within pancreatic tumours first happened. Instead of going forward, the scientists looked back in history using the constant mutation rate of the pancreas as a kind of clock for estimating time. Their conclusion: pancreatic cancer is deadly because diagnosis comes long after the cancer is first triggered.

Early diagnosis is of course the key to cancer treatment. Andrew had symptoms long before going to see his doctor. Like many men he left it later than he should have and his tumours had already breached the protective capsule enclosing his prostate. Fortunately, there are no signs yet of metastasis – when cancer cells spilt off from the primary tumour and take root elsewhere in the body.

His DNA is being analysed by Professor Ros Eeles and her colleagues at the Institute of Cancer Research in London. She has led a monumental effort to check more than 211,000 genetic variants in the DNA of some 25,000 prostate-cancer patients, comparing these single-point mutations to the variants carried by 25,000 healthy men.

Out of this work, published in March 2013, Eeles and her colleagues found 23 new genetic variants associated with an increased risk of prostate cancer, bringing the total number to 78. Individually, these mutations have hardly any effect on risk, but together they exert a significant impact.

There is no mass screening for prostate cancer, the most common cancer of British men, because there is no truly reliable test for the earliest stages of the disease. But this kind of genome research could soon change that.

Eeles hopes the latest result will lead to the development of a simple blood or saliva test to look for inherited genetic variants that collectively raise the risk of prostate cancer in the most vulnerable men. For the first time, scientists would be able to identify the top 1 per cent of men with the highest risk of developing prostate cancer, those who carry nearly five times the risk of the average man.

"These results are the single biggest leap forward in finding the genetic causes of prostate cancer yet made," Eeles says. "They allow us, for the first time, to identify men who have a very high risk of developing prostate cancer during their lifetime through inheritance of multiple-risk genetic variants."

Similar "genome-wide association studies" are underway for breast and ovarian cancer – two other cancers driven by hormones. The aim is to glimpse the genetic deck of cards handed out at birth and to look for those mutations or DNA variants that predispose someone to the risk – however small – of cancer.

Eventually, it will mean that patients receive treatments based on their genes as well as their symptoms. One thing that has become clear from such genetic analysis is that cancers are far more complex than previously supposed – breast cancer, for instance, is now thought to be composed of at least 10 different genetic sub-types.

A personalised approach of targeted therapy could result in many more patients receiving only those drugs that are known to work for their kind of DNA. It could also in the future change the way new anti-cancer drugs are tested.

"In the past, drugs have been developed with large, phase-three clinical trials involving thousands of patients working out what is best for the average," says Alan Ashworth, chief executive of the Institute of Cancer Research. "What we are saying now is to turn this on its head – what is best for the individual patient. It may be that in certain rare cancer types a drug might be considered effective even though there may well never be clinical-trial evidence to prove it," he says.

Andrew's prostate cancer is unlikely to be the result of inherited mutations. None of his male relatives have suffered from the disease, as far as he is aware. But in addition to inherited variations in DNA, scientists are investigating the actual DNA changes taking place within cancer cells over a person's lifetime.

By decoding the entire three billion letters of a cancer cell's genome – several times to make sure there are no mistakes – scientists are able to document the "somatic" mutations that have accumulated during someone's life, rather than the DNA variants they were born with.

These cancer genomes have been likened to an archaeological dig to reveal what has happened to the genetic landscape of a tumour over time. A 90-year-old patient with colorectal cancer, for instance, will typically have twice the number of somatic mutations of a 45-year-old patient with the same kind of tumour. This is because older people accumulate more random somatic mutations, most of which are harmless "passengers".

Cancer geneticists aim to distinguish these passenger mutations from the "driver" mutations that stimulate the growth, development and spread of tumours. Mike Stratton, director of the Sanger Institute, says the insights this will provide could be immeasurable. "It will give us a much more complete idea of what is causing the mutations in our cancers, and what's been causing our cancers in the first place," Stratton says.

For instance, we know cigarette smoking causes specific mutations to the DNA molecule that leads to lung cancer. Very soon it may be possible to distinguish the mutational catalogues caused by different brand of cigarettes because each produces a unique cocktail of about 60 chemicals in the smoke – a measure of the forensic power of genome analysis.

The Sanger has decoded the entire genomes of about 200 breast-cancer tumours, derived from biopsies of individual patients. Worldwide the figure is somewhere between 1,000 and 2,000 whole genomes of breast cancer – a phenomenal amount of data. In the coming years, the number of breast-cancer genomes will increase by 10 or 100 fold.

Serena Nik-Zainal and her colleagues at the Sanger have already found nearly 200,000 mutations in the DNA of cancer cells taken from just 21 patients with breast tumours. She says the mathematical analysis of these genomes is like delving into the archives of a patient's medical records, written in DNA code.

"We can tell which mutations have come on earlier and which have come on later, so we can see additional structure in the development of cancers, and it's far more complex than we had thought," Nik-Zainal says.

One of her most stunning finds was the presence of a band of heavy mutations concentrated on a relatively small region of one of the 23 pairs of chromosomes. It appeared on the computer plot of coloured dots like a heavy downpour falling from the "raincloud" of background mutations hanging overhead.

The researchers called the phenomenon "kataegis" after the Greek for thunderstorm and Nik-Zainal found the effect in 13 out of the 21 breast-cancer patients.

"It looks like a mutational thunderstorm. This is a phenomenon that has never been seen before, and we could not have hoped to see it before we were able to look at the entire mutational catalogue of these cancer genomes," Nik-Zainal said.

Using similar algorithms – sets of mathematical instructions – she found kataegis in other whole genomes, so the phenomenon is not just restricted to breast cancer. This new mutational phenomenon hidden within the genome could be an important trigger for the growth and spread of many other kinds of tumours.

It may be, for instance, that a similar mutational downpour occurred at some point within the prostate gland of Andrew and men like him. Nothing in Andrew's medical history suggests an explanation for his tumours – he says he is as fit as a fiddle – but mutations within his prostate DNA must have been the trigger.

It remains a mystery why some people such as Andrew develop cancer. But the mathematical analysis of his genome, and the DNA of the many thousands like him, may one day lead to an answer. Maths may not provide the cure for cancer, but at least the war on cancer has become more numerate.

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