Cancer Research UK’s Dr Trevor Graham (Barts Cancer Institute) and Dr Andrea Sottoriva (The Institute of Cancer Research) discuss their work on understanding the mathematics of how tumours grow and its implications for treatment.
A cancer forms when a cell in the body goes awry, multiplying out of control to form a tumour. Cancer cells can then break away and travel through the body, setting up home in other organs and creating secondary tumours.
And just as the descendants of early life-forms evolved and changed over time, picking up genetic changes that brought new characteristics allowing them to survive and diversify in different locations, so the descendants of the original cancer cell that gave rise to a tumour evolve and change within the body.
A typically-sized tumour is made up of more cells than there are people on the planet, and researchers now know that cells from different areas of a single tumour have very different alterations in their genetic code.
This sounds like complete chaos. How can we expect to treat cancer effectively – even with newer ‘targeted’ therapies that hit the products of faulty genes – if every cell is different?
In order to find more effective ways to treat the disease, we need to decode some order from the chaos.
The secret diary of a tumour
When we first began to study the patterns of genetic alterations inside human cancers, chaos was exactly what we expected to see. And the first studies initially seemed to back this idea up.
Surprisingly though, the more we looked, the less chaotic the patterns became. In fact, we realised that there was often such striking regularity in the patterns that they might be able to be explained by a simple mathematical formula. And so we set out to see if we could work it out.
In our latest work, published this week in the journal Nature Genetics, we were able to explain how the patterns we found in the chaos of genetic alterations inside a cancer reveal hidden information about how that cancer grew.
In other words, the genetic alterations of cells within the cancer are like a ‘secret diary’, written as the cancer developed. And the mathematical formula we found is akin to a grammatical rule for decoding the words within it. By reading the diary with the help of this rule, we could see exactly how the cancer had grown.
Each time a cell divides to form two new daughter cells, the DNA inside it (which bears its genetic instructions) must be duplicated so that each daughter receives a complete set. Unfortunately, although the process of DNA replication is very accurate, it is not perfect and errors occur during the copying process. These errors are called genetic mutations.
The process of DNA replication is a bit like making multiple photocopies – each successive round of copying contains slight imperfections (or, in the case of DNA, mutations) compared with the original, which build up over time.
As the copies are copied over and over, the imperfections accumulate, introducing ever more defects into the image. And after many rounds of copying, we would expect the image to be very distorted indeed, like the image below.
A similar thing happens as a tumour grows. Cancers start from one cell, that divides to make two cells, and then four cells, then eight, then 16, and so on. Each time the cells divide, they copy their DNA – including any previous errors – meaning that mutations that occur in one division are inherited by all the descendents of that cell. In this way, the DNA from the first cell is progressively distorted as the cancer grows, eventually leading to the genetic chaos we see in tumours.
Our approach was to try and ‘read’ this process in reverse: starting from the end point of a cancer genome with many mutations (a very distorted image that had been photocopied many times) and attempting to decipher the sequence of cell divisions (the order of photocopies and the changes happening with each round of duplication) that would have led to the particular pattern of mutations we observed.
In our study, we used data from 14 different types of cancers that had been collected using a technique called next-generation sequencing, which can tell us two things: first, whether a particular mutation is present in a cancer, and second, the fraction of cells within the cancer that has a given mutation in their DNA.
This technology has produced huge amounts of data, stored in publicly available databases. And to try and make sense of it all, many people have created complex computer programmes that search for patterns in it.
From tiny acorns…
Instead, we decided to take a different approach. We realised that the pattern of mutations within a cancer might make more sense if we looked at them while thinking about how the cancer had grown.
We realised that mutations that happened early in a cancer’s development would be present in lots of cells within the cancer, because these mutations would be inherited by all the daughter cells as the cancer grew. On the other hand, mutations that occurred later would be present in only a few cells.
The interesting twist was when we realised that there should be many more rare mutations in the cancer – each present in only a relatively small number of cells – than common ones. This is because later on in cancer development, when the tumour is larger, there are many more cells dividing and so much more opportunity for mutations to happen.
So we based our mathematical formula on exactly this idea: that a small number of early mutations should be widespread within a cancer, whereas later ones should be more numerous but rare.
The formula we developed perfectly described the pattern of mutations in more than 200 of the cancers we looked at. This meant we knew exactly how these cancers had grown.
And – importantly – it shows that the pattern of mutations in a cancer follows something known as a ‘power-law’. Power-laws are found in all kinds of natural systems. Consider earthquakes, for example. There are almost unnoticeably small seismic events happening almost constantly around the globe, but cataclysmic earthquakes only happen once a decade or so. It turns out that average time between different types of earthquakes obeys a power law.
In cancer, we found that the way the cancer grows means that mutations present in large numbers of cells (large events) were rare, whereas mutations present in only a few cells (small events) were commonplace. In other words, this suggested we could expect to find a power-law underlying cancer’s growth. And that’s exactly what we did find.
We’re excited to have found a natural law of cancer growth that reveals striking simplicity in the apparent chaos of a cancer genome. We think this work is important and is a first entry in what we hope will become a ‘mathematical rulebook for cancer’, that will serve to simplify and improve our understanding of the disease.
Our work draws upon ideas from an area of scientific research called ‘population genetics’. For the last few decades, population geneticists have been developing mathematical tools to make sense of complex genetic data from large groups of organisms including humans, animals, plants and even micro-organisms such as bacteria. In the future, we’re eager to apply more of the powerful ideas from population genetics to better understand cancer development.
Ultimately this will help us to work out further rules, with the end goal of finding something that can help doctors to predict how cancers will grow and change, treating them more effectively in the future.
Obviously there’s a long road ahead. But now, as we’re starting to find order among the chaos inside a tumour, we’re also finding new clues to better target the disease which we hope will make a difference to the people who suffer from it.
A version of this piece was also published on The Conversation.