Concluding Remarks on Wound Healing

Although we have said it before (Murray et al. 1998) it bears repeating to ask why we use mathematics to study something so complicated and badly understood as wound healing. We argue that mathematics must be used if we hope to truly convert an understanding of the underlying mechanisms into a predictive science. Mathematics is required to bridge the gap between the level on which most of our knowledge is accumulating (cellular and below) and the macroscopic level of the scar itself which is of primary concern to the surgeon and the patient. A mathematical approach allows one to explore the logic of wound healing. Even if the mechanisms were well understood—and they certainly are not at this stage—mathematics would be required to explore the consequences of manipulating the various parameters associated with any particular wound management scenario. The number of options that are fast becoming available to wound managers will become overwhelming unless we can find a way to simulate particular treatment protocols before applying them in practice.

We should also remember that an important point arising from the models is that a scar contains its own history. Consider an engineering analogy (Murray et al. 1998) of our role in managing the wound healing process. It is one thing to suggest that a bridge requires a thousand tons of steel, that any less will result in too weak a structure, and any more will result in excessive rigidity. It is quite another matter to instruct the workers on how best to put the pieces together. It is conceivable that the cells involved in tissue repair have enough 'expertise' that given the right set of ingredients and initial instructions they could be persuaded to repair a deep skin wound with a result that looked more like skin than scar tissue. This is perhaps the hope of those who are searching for the optimal cocktail of growth factors to apply topically to a wound. However, it seems to us very likely that the global effect of all this sophisticated cellular activity would be critically sensitive to the sequence of events occurring during tissue repair. As managers we should concern ourselves with how to take advantage of the limited opportunities we have for communicating with the workforce so as to direct the wound healing process towards an acceptable end-product. This may sound rather philosophical, but even a cursory look at the theories of scar formation in the literature reveal a fixation on simplistic explanations. For example, it has variously been suggested that pathological scars result from defects in collagen synthesis or in collagen degradation.

We suggest that there need be no misfunctioning of any particular subprocess to explain scar formation. Tissue repair is, after all, fundamentally different from development. The body has only a limited set of 'rules' with which it must confront an unlimited number of possible repair 'problems.'

We are without question a long way from being able to reliably simulate actual wounds. Not only is the active cellular control of the key processes poorly understood, but also wounds are difficult to reproduce, with similar wounds on different parts of the same body, healing at different rates and with different results. Despite these limitations, we argue that exploring the logic of wound healing is worthwhile even in our present state of knowledge. It allows us to take a hypothetical mechanism and examine its consequences in the form of a mathematical model, making predictions and suggesting experiments that would verify or invalidate the model; even the latter casts light on the biology. Indeed, the very process of constructing a mathematical model can be useful in its own right. Not only must we commit to a particular mechanism, but we are also forced to consider what is truly essential to the process of wound healing, the key players (variables) and the key processes by which they evolve. We are thus involved in constructing frameworks on which we can hang our understanding of wound healing. The equations, the mathematical analysis and the numerical simulations that follow serve to reveal quantitatively as well as qualitatively the consequences of that logical structure.

The use of mathematical model mechanisms for the study of wound healing has already shed light on several hitherto poorly understood topics. The rapid increase in biological technology has led to more sophisticated experiments and such research builds on advances in our understanding of the cell-matrix interactions, the use of growth factors and biomaterials, more powerful computers and advanced cell culturing techniques. The experimental work such as that of Green (1991) mentioned above and the techniques by Bertolami et al. (1991) on skin substitutes are just two examples. The available information now lets us try and realistically relate microscopic and macroscopic phenomena with mathematical models. The work of Cook (1995) along these lines provides a basis for studying the major problem of granulation tissue and scar formation during wound contraction and matrix remodelling. Pathologic scars are related to abnormal structure of the ECM (Dunn et al. 1985). The work of Dale et al. (1995) and Olsen et al. (1995) is particularly relevant here. These models also let us begin to seriously address the surgical issues of wound orientation with respect to tension lines, wound shape and geometry so as to minimize scarring. We believe that the approach of building models from the microscopic (cell level) to determine the macroscopic models such as we have done here is a fruitful one. Realistic models must eventually incorporate finite strains which makes the development of appropriate constitutive equations for biomaterials particularly pressing. Crucial to this are microscopic and macroscopic measurements of cell traction within biological tissue, some of which has been done by, for example, Delvoye et al. (1991), Ferrenq et al. (1997) and Tranqui and Tracqui (2000).

None of the individual models we have discussed and not even all of them put together could be considered a complete model of dermal wound healing, even if we think only in terms of wound contraction and scar formation. However, each model has shed light on different aspects of the process and we can now say what must be the most important elements of a complete model. These studies have served to highlight where our knowledge is deficient and to suggest directions in which fruitful experimentation might lead us. Indeed, a critical test of these theoretical constructs is in their impact on the experimental community. The field has now achieved some level of maturity and we believe that future dialogue between experimentalists and applied mathematicians will lead us most rapidly towards the goal of scarless wound healing.

11. Growth and Control of Brain Tumours

Brief Historical Perspective on Brain Surgery

Surgery on the brain or skull has an incredibly long history and is arguably the most ancient of surgical interventions. The books of articles edited by Greenblatt (1997) and Walker (1951) make fascinating reading. There is an early description of how to treat head wounds in the Edwin Smith Surgical Papyrus of ancient Egypt (about 2500 BC) in which trepanning, or trephining, is described as a procedure. Trephining is the surgical removal of a piece of bone from the skull. Numerous skulls have been found showing healed craniotomies; the procedure was certainly far from always being fatal and the large number of such healed craniotomies suggests some success.

Verano and his colleagues (see Verano et al. 1999, 2000 and Arriaza et al. 1998 and other references there) discuss interesting examples, from archaeological finds, of trephination, and other major surgeries such as foot amputation, by the coastal cultures of Peru and the Incas. There is a remarkable photograph in Arriaza et al. (1998) of an Inca cranium (from the archaeological museum in Cuzco) showing four trephinations which were clearly well healed while the patient was alive. The trephinations on this skull are all of the order of 5 cm in diameter.

There are numerous pictures and woodcuts from the mediaeval period (see also the historical discussion on surgery and wound healing in Chapter 9) showing surgical operations on the head. The procedure for carrying out such trephining varies widely. Figure 11.1(a) is from a mediaeval manuscript showing how it might have been carried out while Figure 11.1(b) is from another mediaeval manuscript showing the end of a succesful completion of the surgery (albeit with a somewhat dubious and unhappy looking patient). Figure 11.1(c) is a caricature example of a late mediaeval operation, this time being carried out by a group of animal surgeons which indicates what people thought of surgeons at the time. As mentioned in the chapter on epidermal wound healing, Galen, who clearly carried out trephining, was very practical in his advice such as noting that the 'disadvantage of using a gouge is that the head is shaken vigorously by the hammer strokes.'

Richard of Parma in his Practica Chirugiae (Practice of Surgery) of 1170 (see Valenstein 1997) writes: 'For mania or melancholy a cruciate incision is made in the top of the head and the cranium is penetrated to permit the noxious material to exhale to the outside. The patient is held in chains and the wound is treated, as above.' The classic picture by the late 15th and early 16th century enigmatic Hieronymous Bosch called The Cure of Folly (or Madness) or The Stone Operation in the Prado Museum in Madrid

Figure 11.1. (a) Mediaeval illustration of the start of a skull operation. (b) This is from a late mediaeval manuscript showing post-surgery suturing. (c) Brain surgery being carried out by a number of animal 'surgeons.' It highlights the low opinion people had of surgeons in mediaeval times. (Courtesy of the Wellcome Trust Medical Library, London and reproduced with permission)

Figure 11.1. (a) Mediaeval illustration of the start of a skull operation. (b) This is from a late mediaeval manuscript showing post-surgery suturing. (c) Brain surgery being carried out by a number of animal 'surgeons.' It highlights the low opinion people had of surgeons in mediaeval times. (Courtesy of the Wellcome Trust Medical Library, London and reproduced with permission)

shows a man making an incision on the scalp of a seated man. The inscription (translated by Cinotto 1969) has been translated as 'Master, dig out the stones of folly, my name is "castrated dachshund."' At the time a 'castrated dachshund' was frequently the name used for a simpleton. Madness was often believed to be caused by stones in the head. Art historians and others have discussed and disagreed about the interpretation of this painting for a very long time; some, for example, say it ridicules the itinerant mediaeval charlatans who went around 'curing' madness and other ailments by trephining while others say it is an allegory of human gullibility and stupidity which is so common in Bosch's paintings. The classic book Anatomy of Melancholy by Robert Burton (1652) includes 'tis not amiss to bore the skull with an instrument to let out the fuliginous vapours ... Guierius curted a nobleman in Savoy by boring alone, leaving the hole open a month together by means of which, after two years melancholy and madness he was delivered.'

The practice of trephining declined from the beginning of the 19th century when the surgical operation was moved into the hospitals because the mortality went up dramatically since such hospitals were infection paradises at the time. (Even now infection in hospitals is still a major killer, even if the prognosis of survival is much greater.)

Trephination by indigenous peoples has been used (and still is today) for a variety of problems, such as insanity, depression, behavioural disorders (lobotomies are still carried out by professional surgeons in many hospitals) to relieve the results of head injuries from blows to the head (such as depressed fractures) or falls from trees or simply from banging their head on a door lintel. Margetts (1967) and in particular the interesting article by Furnas et al. (1985) (see earlier references there) describe (with many photographs of actual operations) in fascinating medical detail craniotomies regularly performed by traditional craniotomists of the Kisii tribe in Kenya. The craniotomist (called an omobari) is highly respected and very skillful. They use a variety of herbs and drugs to aid healing and prevent infection. The operations are often carried out in the open. They use a series of instruments to scrape through the skull, roughly a 5 x 5 cm2 patch, until a very thin layer is left. They then use a fine pick to puncture this last membrane thereby exposing the dura. Some, however, also use a drill or hacksaw. The aim of the treatment can be, for example, to remove bone splinters (resulting from a blow to the head) or to relieve depression or severe headaches. It appears that the mortality rate is low and patients seem satisfied with the results.

How To Reduce Acne Scarring

How To Reduce Acne Scarring

Acne is a name that is famous in its own right, but for all of the wrong reasons. Most teenagers know, and dread, the very word, as it so prevalently wrecks havoc on their faces throughout their adolescent years.

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