Subsections
• Mathematics helps to clarify the underlying mechanisms; this is because the process of modelling makes us concentrate on separating the essential from the inessential.
• A model can be used to investigate situations that are not easily amenable to experiment.
• Mathematics is valuable in testing out ideas.
• Mathematics is a concise but powerful language that allows the `hidden' similarities between different systems to be discovered.

  
Mathematics in Biology and Medicine

Not many people know that the application of mathematics in Biology and Medicine provides a wealth of opportunities for mathematical modellers. This developing interface is both exciting and stimulating. All these fields are enriched by the interaction.

Modelling provides another way of looking at biological systems and also at medical phenomena. There are clearly a number of advantages to this approach:

  
Mathematics helps to clarify the underlying mechanisms; this is because the process of modelling makes us concentrate on separating the essential from the inessential.

Transporting epithelia are cells which form in layers in the body. Their primary function is to pump fluid from one side of the cell to the other. In the kidneys, epithelial cells reabsorb the filtered blood. The `standing gradient hypothesis' to describe the pumping mechanism, when put into the quantitative form of a mathematical model, did not stand up to careful scrutiny. The predictions were not consistent with experiments. Alternative mechanisms are now under investigation by kidney physiologists.

  
A model can be used to investigate situations that are not easily amenable to experiment.

We at QUT are doing research on the retinotopic nature of the mapping of the visual field onto human visual cortex. What we want to know is how much of the visual cortex is devoted to each part of the visual field. We cannot do invasive experiments on the human brain. However, what we can do is to create a 3-dimensional reconstruction of the brain, using magnetic-resonance imaging. We are measuring electrical impulses on the scalp, then using a mathematical model to locate which part of the brain is active when a light is presented in the visual field.

A good model of a fishery can be used as a predictive tool. We can attempt to project what will happen to fish populations if we periodically close the fishery or put quotas on catches. The International Whaling Commission already uses a mathematical model to examine the effect of harvesting the baleen whale population. Considerable effort has been put into a model of the prawn populations around Fraser Island in Queensland.

  
Mathematics is valuable in testing out ideas.

Experiments have shown that in isolated tumours, the cells move from the outside towards the centre. They also show signs of being able to exhibit active movement. Recently we have developed a model of this system. We conclude that the cells are moving due to a gradient of pressure. They move from the outer region, where there is plenty of oxygen and the pressure is high due to cell division, to the centre where the pressure is low. They do exhibit some active migration, but then they are crawling against the pressure tide. In this case, the mathematicians have obtained a result that is contrary to intuition, but is consistent with the experiments.

  
Mathematics is a concise but powerful language that allows the `hidden' similarities between different systems to be discovered.

White cells packing into arthritic joints, the attraction of blood vessels to and into a tumour so that it gets a good supply of nutrients and oxygen and can grow explosively, the in-growth of blood vessels in a wound and the implantation of a trophblast into the wall of the uterus are all situations that can be described by the same type of mathematical model. In each case, cells are reacting to a chemical signal and are migrating towards the source. The power of mathematical modelling is that it provides a quantitative framework to examine all of these phenomena. Mathematical predictions obtained in one area may well have applicability in another.

When I am modelling I am reminded of the quote:

If the Lord Almighty had consulted me before embarking on creation I should have recommended something simpler.
Alphonso X (Alphonso the Wise), 1221-1284.

and I try to keep the models simple. But I have to admit that I am rather pleased that the fields of Biology and Medicine are so complex; they provide an infinite opportunity for the mathematical modeller.