For so many people the thoughts of studying maths are inevitably filled with memories of stifled yawns in classrooms, struggles to
work out if x really does equal y, and wondering what could possibly be the point of trigonometry?
It is a subject that is universally taught in our schools and yet is possibly the one most people will say is least used once they leave,
or almost certainly the one they forget the most about - when was the last time, for instance, you felt the need to determine the
stationary points on a curve? And would you have the slightest idea of where to start if you did suddenly have the urge?
Maths, it is fair to say, has had a remarkably bad press. Although, as Professor Mark Chaplain admits, the subject itself has to
take a good share of the blame.
'Maths is not a subject where you can just dip into it and extract something and look at that on its own,' said Mark,
Head of the Division of Mathematics at Dundee. 'Maths is a subject that builds one layer on top of another.
All of which, unfortunately, makes it impossible to teach all of the exciting stuff at Standard Grade!'
But the 'exciting stuff' is there, and increasingly it is playing a vital role in unlocking the puzzles of some of
our major diseases, cancer chief among them.
Mathematical models which can predict how tumours form and how they will behave have become an
advanced tool for mapping out how cancer affects our bodies.
'The models we have developed can, and have, charted all the stages of cancer,' said Mark. 'The formation of
tumours, how they develop and grow, the properties they possess, how aggressive they might be, all of these
things can be predicted using mathematical modelling.
'A good comparison of how it works is the weather forecast. When 'Heather the Weather' presents the television
weather report, what you are seeing is the end result of a very detailed piece of modelling, where things like
air pressure, temperature, wind speeds and so on are the factors which are taken into account to develop a
final picture.
'Our modelling does the same for a biological situation. Unfortunately for us, a biological
system is far more complex than the weather system.'
It is also, of course, far more important to get it right. But the value of doing so is huge.
A successful model has the potential to replace years and years of practical work -
and associated costs - in predicting how tumours will behave. This, in turn, gives
clinicians and researchers valuable insights into the best ways to treat them.
'Maths underpins everything we see around us,' said Mark. 'You can make a model
of anything using maths. The real trick is to make a very good and predictive model
and do it well, which is where it gets complicated. But a good model will give new
insights into lots of things.
'I got into this after reading in biological texts that cancerous tissue behaved differently
from normal tissue. If that was the case then there had to be a way of describing that using
maths and modelling.
'What we do in making a mathematical model is we translate all the biology into a mathematical
description. You then fill in that model with experimental data and from that, using analytical and
computational techniques, you gain qualitative and quantitative results i.e. predictions.
'One of the areas we have looked at and had some impact in is angiogenesis, the formation of
blood vessels. Our modelling has shown how the structure of these blood vessel networks is very
important, especially when you go on to look at how treatments such as chemotherapy are delivered.
'The blood vessels are vital for the cancer to survive and grow, but they are also an essential pathway for
the delivery of drugs to treat the tumour, so it is very much a double-edged sword. But by examining them through a very efficient model, we can determine the best timings of when to get the drugs in there, and then how to address the problem of closing off the pathway.
'Our work is giving an understanding of more efficient methods of drug delivery. This has been applied to radiotherapy treatment as well, where we have shown some real differences between current treatment methods. This, among other things, has given a possible explanation of why some women suffer a recurrence of breast cancers five years after treatment.
'Using modelling effectively means we are maximising the chances of getting it right when it comes to looking at cancer in a clinical setting. If you were to try and find these things out by doing physical experiments, it would literally take years, cost a lot of money and involve physical experiments. Using the mathematical models, you can remove a lot of that work.'
Mark has been a leading figure in this field for more than 20 years, and heads a Maths division at Dundee which has a worldwide reputation and now enjoys links with the likes of Harvard and Vanderbilt in the USA and is engaged in major European research networks.
He has witnessed a sea-change in the level of interest directed towards maths potential as a tool to be used in medical research.
'I did my PhD thesis in Dundee in the late 1980s and there was nobody else doing this sort of work in maths related to cancer. It was a lonely field to be working in - I would turn up at conferences and be the only person speaking in this area.
'I would turn up at the medical school here at Ninewells and I would describe the reception to what I was saying as something like polite bemusement! But the stuff I did in my PhD has subsequently proved to be practical and extremely useful, which has the habit of changing perceptions.
'The whole thing has turned on its head. More and more biologists now are knocking on our doors to start using this. The position of the research councils has also changed entirely. They are now recognising the value of mathematical modelling applied to biomedical sciences and the exciting opportunities to be had in interdisciplinary research.
'Particularly over the last five years or so there has been a big increase in interest in this area, which has led to more and more people doing it. That means there�s a lot more competition in the field, with a lot more groups now working in the maths/cancer modelling in the UK and the rest of the world.
'But we are still in a good position in Dundee. We have been doing this for a long time and we have some of the best models for looking at certain types of cancers.
'The increased interest comes because cancer is like all of biology in microcosm. All of the things that happen across the wide spectrum of biology - generation of cells, growth and mutations, evolutionary selection - these all happen in cancer, only much faster. Cancer effectively speeds up the timescale of nature, and because of that you can look at it and model a lot of different things.
'In cancers you see mutations upon mutations, lots of different things happening and doing it relatively very quickly. That gives us a lot to look at.'
It has also had a rejuvenating effect of maths itself as a whole new field of investigation has opened up.
'The problems associated with the biomedical sciences have opened up entirely new areas for the mathematical sciences - pure maths, applied maths and statistics,' Mark explained. 'New tools, new techniques, new theories are all being developed and brought into play as we try to gain an understanding of these biological problems.'
Allied to huge computing power, these factors are knitting together to find more and more of the answers that are needed if we are to solve the cancer test.
'All the time we are building better models and getting closer to delivering better and more informed systems for clinicians and giving a genuine insight into some very, very complicated systems,' said Mark. 'It is going to take a wide variety of factors to beat cancer - and maths is right there among them.'