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CUT yourself. You bleed and in time the wound will often heal without even a scar. But behind this everyday process there is a complex sequence of biochemical events. All round the world researchers are trying to work out exactly how wounds heal so that they can develop treatments to speed up the process and prevent scarring. Now they have been joined by mathematical biologists. Armed only with their stock in trade, equations and computers, they are making important contributions to understanding the healing process. Modern mathematical biology is about 25 years old but its devotees only turned their hand to wound healing in the last five years. Its aim is to take medical findings and make new predictions based on mathematical models and computer analysis. By feeding these predictions back into the real world of experimental research, mathematical biologists hope to speed up our understanding of wound healing by suggesting the most fruitful lines of enquiry.

Clean cut

In shallow wounds, for example, the key issue is speed of healing. Although such wounds normally heal efficiently without scarring, simulating healing can test ideas on what factors affect the rate of healing. Deeper wounds, on the other hand, leave scars, so researchers are more interested in the quality of healing than the speed. Scar formation is influenced by biological molecules. Mathematical models are helping to identify which molecules play the largest role and how the balance might be tipped to reduce scarring. Computer simulations are also being used to understand why deep wounds sometimes contract excessively when they heal.

We have all seen dogs licking their wounds. As well as cleaning the cut, the saliva contains molecules called growth factors – a family of about 50 proteins that regulate cell behaviour. Growth factors control every aspect of wound repair. Initially they come from immune cells in the blood which cluster around a new wound. But as it heals, growth factors are produced by all the different types of cell in a wound. Because their influence on the speed and quality of wound healing is so great, understanding their precise role is crucial if we want to intervene in wound repair.

Mathematical biology is a very efficient way to study how growth factors work. Mathematicians can make specific predictions based on what they already know about how particular growth factors work. Their basic tools are mathematical models that represent what happens in terms of equations. Usually they solve these equations using computer simulation, and then other researchers can test the new predictions in real wounds.

For example, fast effective healing of the cornea is essential to the success of common forms of eye surgery, including radial keratotomy to correct myopia. For some time my group at the University of Warwick has been working with Philip Maini’s team at the University of Oxford to find out how the surface layer of the cornea heals. Growth factors play a major role here, speeding up cell division and giving a pool of new cells to fill the wound. What is not clear is where most of these growth factors come from. Likely candidates are the tear film that lubricates our eyes, the corneal cells themselves and underlying tissue.

We are trying to solve this mystery using a type of equation known as a conservation law to describe both growth rate and the influence of various growth factors.

Net increase of cells = change due to cell movement + change due to cell division and death

Net increase of growth factors = change due to diffusion of growth factors + change due to growth factor production and breakdown

At a basic level, conservation laws are just common sense. For example, you could calculate the rate at which a population is increasing if you knew the numbers of people entering and leaving the country and the numbers of births and deaths. But no one knows the exact details of cell movement, division and death and growth factor levels during wound healing.

Natural healing

The art of modelling comes in reflecting existing biological knowledge in terms of mathematical formulae – the term for cell movement, for example, must take account of the curvature of the eye. Once this has been done, variables such as the source of the growth factors can be substituted into the equation. So the model becomes a mathematical experiment. Some possible sources of growth factor fit simple observation, others can be ruled out – the tear film, for example, does not seem to contain high enough levels of growth factor to account for normal healing. The computer simulations of healing will suggest specific experiments and observations that will help to distinguish between the possible sources.

Once we know which growth factors are involved and where they come from, doctors should be able to speed up healing in their patients by adding extra growth factor to a corneal wound. Experiments by Roger Beuerman and colleagues at Louisiana State University in the early 1990s have already proved that this works. Mathematical modelling will tell us the quantity and type of growth factors to apply to wounds of different shapes and sizes and the best time to apply them. Our simulations, for example, suggest healing has an upper speed limit. And we can predict the minimum amount of growth factor that will give this optimum level of healing.

If a wound is superficial, our aim is to accelerate repair. For deep wounds, however, it is more important to prevent scarring. The advantages of this are twofold. As well as looking ugly, scar tissue is also much weaker than normal skin. Growth factors play a central role in scarring and mathematicians are simulating scar formulation using models with various combinations and amounts of growth factors. A clue to which growth hormones might promote healing without leaving a blemish comes from seemingly unrelated experiments, which began almost a decade ago.

Fetal surgery

In the mid-1980s Scott Adzick and his colleagues at the University of California in San Francisco realised that advances in biotechnology made fetal surgery a real possibility. The implication was that they might eventually be able to correct conditions such as diaphragmatic hernia which poses no problems for a fetus but can kill a newborn through suffocation. During their initial experiments on animals, Adzick’s team made an important discovery. Fetal wounds heal without scarring.

By 1990 Adzick’s group had performed 18 operations on human fetuses. Around this time Mark Ferguson and his team from the University of Manchester began a series of experiments to discover how fetal wounds heal differently from those of adults. Their work continues, but so far they have shown significant differences in the concentration of some growth factors in fetal and adult wounds. And they have been able to reduce scarring in wounded adult animals by tipping the balance of various growth factors to make them more like fetal wounds.

These experiments have pinpointed growth factors that regulate scar formation. Now the mathematicians are stepping in to fine-tune the findings. Using computer simulations, Julian Cook’s group at UCLA and my team at Warwick together with Maini’s at Oxford are developing a detailed understanding of the interaction between different growth factors and their exact function in controlling how new tissue forms. The basic idea is the same as with corneal healing – equations express a combination of biological facts and theories and then computer simulations test out the hypotheses. But scar formation is a much more complex process than corneal healing and the mathematical models reflect this.

Both normal skin and scar tissue are built around a network of protein fibres. In normal skin these collagen fibres form a sort of basket-weave pattern, whereas in scar tissue they are thinner and more densely packed with pronounced alignment. During repair, cells within the wound secrete these fibres and we are now using computer simulations to predict how the balance of growth factors affects the outcome. Trying out ideas on a computer rather than on a patient means that the analysis can focus on a single aspect of fibre secretion and ignore peripheral influences that might cloud the picture. How, for example, do different growth factors regulate one another’s action, and which interactions are the most fundamental to the fibrous structure within a scar? This research is at an early stage but it might lead to new ways of inhibiting scarring.

Already, experiments in animals have shown that scarring can be reduced by chemical intervention during healing. Ferguson’s team in Manchester has done this by injecting wounds in rats with a chemical that neutralises one of the growth factors. Now my group is using mathematical modelling to predict how this treatment might best be used in humans. Such simulations are much faster and cheaper than experiments with real wounds. They can, for example, suggest optimum doses and the best time to intervene. Predictions can then be checked with just a few studies on real wounds.

Restricted movement

Another important factor in the healing of deep wounds is contraction. While individual cells in a wound exert only a very weak force, acting together, like members of a tug-of-war team, they are able to pull the edges of a wound together. This is essential for normal repair but if the wound is on a limb too much contraction can lead to permanent and painful restriction of movement.

Mathematicians have been interested in the forces that control contraction for over a decade. In an intensive research programme during the 1980s, James Murray and colleagues at the University of Oxford and the University of Washington in Seattle, used mathematical modelling to show that many aspects of fetal development can be explained by tension forces exerted by cells in the embryo. In 1988, Murray realised that these mechanisms are similar to those in wounds and began working with Robert Tranquillo from the University of Minnesota to extend his models to wound contraction. By combining the original simulations with data from experiments they created a new model that is now used around the world.

Much of the experimental data about how contraction works come from studies with artificial wounds called collagen lattices, developed in 1979 by researchers at the Massachusetts Institute of Technology and Harvard University. About the size of a pound coin, these are made of protein fibres similar to those found in a real wound. When wound cells are added to the lattice it shrinks visibly. By adding individual growth factors to the lattice the researchers can test how they influence contraction. Data from such experiments give a mathematical biologist precise information that can be used to tweak a single term in an equation.

Modelling has been particularly useful in understanding wound contraction in humans. Most experiments on real wounds are done using animals whose skin is only loosely attached to the underlying tissue, whereas human skin is held tightly in place by a special subcutaneous muscle. Wound contraction in animals can shrink the skin to a fraction of its original size – something that does not happen in human patients. This is where mathematics comes in.

Mathematicians can extrapolate from animals to humans simply by adding a term into the equation that allows for the firmer attachment of human skin. In Murray and Tranquillo’s model, for example, the attachment between skin and underlying tissue is represented by numerous tiny springs. The tension in these springs resists the contractile pull of cells within the wound and their strength is all that distinguishes human and animal skin. Other researchers led by Frank Arnold and George Cherry at the Churchill Hospital in Oxford, are using predictions from similar models to guide their work on real wounds. They believe that within a few years their work may offer new treatments for chronic ulcers, and other wounds that fail to close up.

In the Warwick-Oxford teams we are also studying abnormal wound repair. We have modelled two poorly understood complications of wound healing. In both cases more new tissue is produced than is needed simply to fill the wound. Most commonly, particularly with burns, the result is a bulging scar or hypertrophic scar. Occasionally, the new tissue spreads out beyond the edges of the wound to create what looks like a large angry swelling – a keloid scar.

By changing the parameters of our models we can simulate the formation of hypertrophic and keloid scars. There seems to be a critical level for making new growth factors, which in normal wounds is delicately balanced with the breakdown of existing growth factors. Tip the balance to overproduction and healing becomes abnormal. If experiments confirm this we might have the first clues to preventing abnormal repair. What makes studying wounds with mathematics so exciting, is the possibility of one day improving on nature. If our collaborations with medical researchers are successful, scarring could be a thing of the past. Mathematics might help produce a fast effective treatment in the form of a simple gel.

A formula for health

THE mathematicians are muscling in on medicine. Cancer is one area where their models can make a big difference. Tumours start when a single cell mutates giving it a growth advantage over neighbouring cells. Several mathematicians have used models to study the growth implications of different types of mutation. This is part of a huge research effort to understand the genetic origins of cancer and eventually generate novel therapies.

Once they measure more than a few millimetres across, tumours have to generate their own blood supply to continue growing. The cells therefore produce a chemical which attracts blood capillaries, just as the scent of a flower attracts insects. Mark Chaplain at the University of Bath has used mathematical models to calculate the range and time scale of this attraction process. His simulations given an important insight into what is happening at this critical stage of tumour growth.

Mathematical modelling of physiological disorders was pioneered in the 1970s by Michael Mackey and colleagues at McGill University in Montreal and is now widely used. In the Cheyne-Stokes respiratory condition, for example, modelling has revealed a crucial factor causing the irregular breathing patterns and recurrent breathlessness associated with the condition. Normally there is a delay of about 10 seconds between the oxygenation of the blood in the lungs and recognition of this by cells in the brain. Computer simulations show that a small increase in this delay time causes a sudden switch to an irregular breathing pattern. Now physiologists are trying to discover what causes that increase and how to prevent it.

Another big area of interest for mathematical biologists is HIV. Here the key issue is to explain the long time lag – 10 years on average – between infection with the virus and the development of AIDS. At the University of Oxford, Martin Nowak’s computer simulations suggest that the lag reflects the immune system’s ability to control HIV until the number of mutant virus strains reaches a critical level beyond which the immune system is overwhelmed.

Alan Perelson at the Los Alamos National Laboratory has a different model. He predicts that during the lag phase the immune system eliminates fast-replicating viral strains but that some slow-replicating strains slip through this net and these strains ultimately leads to AIDS. Clinical researchers are now reassessing their data in the hope that one of these theories may fit with the evidence from their patients.