Annals of the Royal College of Surgeons of England (1976) vol 58
Mathematics and the J
Crank
DSC
FIIlStP
surgeon
FIMA
Professor and Head of the School of Mathematical Studies, Brunel University, Uxbridge, Middlesex
Summary The surgeon uses elementary mathematics just as much as any other educated layman. In his professional life, however, much of the knowledge and skill on which he relies has had a mathematical strand in its development, possibly woven into the supporting disciplines such as physics, chemistry, biology, and bioengineering. The values and limitations of mathematical models are examined briefly in the general medical field and particularly in relation to the surgeon. Arithmetic and statistics are usually regarded as the most immediately useful parts of mathematics. Examples are cited, however, of medical postgraduate work which uses other highly advanced mathematical techniques. The place of mathematics in postgraduate and postexperience teaching courses is touched on. The role of a mathematical consultant in the medical team is discussed. Introduction It could be argued that Brunel University owes its name to a igth-century application of mathematics in surgery. In I 843 I K Brunel, the famous engineer, swallowed a half-sovereign which lodged in his windpipe. A tracheotomy operation performed by the eminent surgeon Sir Benjamin Brodie proved unsuccessful. Brunel himself then designed a simple apparatus to which he was strapped and spun rapidly head over heels. At the second attempt, under the action of the so-called centrifugal force, the coin was dislodged and dropped from his mouth. Brunel lived and a university now bears his name. It seems natural to start by commenting on the title to which the organizers of the symposium invited me to speak. Although nerhans lacking in literary comnatibility, it does indicate precisely the relationshin we wish to consider. One alternative. 'The mathematician and ihe surgeon', would have concentrated
attention on a more personal relationship than is intended and would have implied that all the mathematics of value to the surgeon comes to him through mathematicians, which is certainly not true. On the other hand, 'Mathematics and surgery' would overlook the fact that a surgeon's use of mathematics is not confined to the operating theatre and would suggest an academic discussion of the intersecting areas of the two specialisms. First of all, a surgeon is an intelligent layman and as such must be able to count, perform elementary arithmetic, and even understand simple graphs and statistics. These basic skills and concepts are expected of every educated person. To quote Professor Parkhousel, 'This is comparable to the fact that we, as doctors, would expect any mathematician to have a general knowledge of hygiene, first-aid, child care and development, nutrition, emergency resuscitation, and so forth'. It is difficult to make a case for more than simple mathematical techniques for the man in the street. As a graduate in medicine, however, the surgeon will have studied more mathematics, usually included in physics and chemistry, biology, biochemistry, and perhaps even anatomy and bioengineering. A practising surgeon is caught up in a web of mathematics, for the most part invisible and of which it could be argued, and is, that he need know nothing. There are obvious parallels with people who drive cars without understanding anything aboiut the internal combustion engine or cheerfully watch television without understanding the electronic circuitry. If I may quote Professor Cruickshank2, 'One is inclined to forget that as recently as one huindred years ago empiricism dominated the practice of the "art" of medicine, and "science" was just beginning to influence the thinking of the medical practitioner. It is the ranid development of physics and chemistry, under-
Delivered at the Annual Meeting of Fellows and Members on ioth December I975 as part of a symposium on 'Teaching postgraduate surgery'
Mathematics and the surgeon
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pinned by mathematics, that has made all our advances possible: the development of the microscope; the discovery and use of radiation for diagnosis and therapy; the explosion of clinical laboratory investigations (doubling in number every five years) for diagnosis and prognosis; and the enormous expansion of drug therapy, where io-6 mg of a substance can radically alter cellular function and reproduction-all these virtually within the lifetime of most of us-are testimony to what we owe to the application of mathematics to the physical and biological sciences'. At every surgeon's elbow is not only a nurse, but a mathematician. It is at postgraduate level, however, whether taught courses, seminars, conferences, or research, that the surgeon is most likely to find a need to extend his knowledge of mathematics. We shall return to a consideration of postgraduate work later, but sufficient to say here that even a casual reading of medical research journals reveals how much of the work depends on mathematical techniques. Mathematical models Before we look at the special interest mathematics may have for the siurgeon it is worth while to examine a little more carefully the process of applying mathematics to any practical problem of real life. Somehow we have to express
our real problem in such a way that mathematical techniques which are essentially abstract can be brought to bear. The process has been described by Professor Synge in the following way: 'The use of applied mathematics in its relation to a physical problem involves three stages; a dive from the world of reality into the world of mathematics; a swim in the world of mathematics; a climb from the world of mathematics into the world of reality, carrying a prediction in our teeth'. Figure gives an artist's impression of this diving and swimi
ming.
In the mathematical world we use mathematical language. Both our professions are guilty of using specialist language or jargon and forgetting that this is probably not understood by those outside the particular 'field. Professor Aitchison3 has identified a 'newmath syndrome' with the following symptoms: Algebritis-an obsessive fear of operations, particularly of the algebraic type. Binary paralexia-an inability to manipulate and coordinate mathematical symbols. Calculus deficiency-a failure to integrate properly. They could easily be accepted as genuine diseases by an innocent reader. Mathematicians use diagrams, graphs, and dy . As an example, a symbols-x, y, 0, dx
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FIG. 2
problem referring to a seal sitting on an ice floe could well occur in school mathematics, leading to the question, 'If the ice floe tips up, how long does it take for the seal to enter the water?' Any sixth-fonrmer would draw a diagram and one might expect Figure 2, but he will instead draw Figure 3 and expect everybody to understand that this represents a seal on an ice floe. In fact, he has unconsciously performed a process that we call abstraction, or the building of a mathematical model.
FIG. 3
portant. We make assumptions sufficiently simple to reduce the problem to one that we can handle mathematically but not so simple that our results will be meaningless. Whether a particular assumption is acceptable or not depends on the use to which a given model is to be put. So mathematical modelling is a skilful art. I venture to suggest that there is a close parallel between this modelling and the medical treatment of a patient. Figure 4 illustrates this analogy. In the mathematical case we proceed from the real problem, having decided what are the really vital features, to the model and thence to a suitable mathematical treatment of our equations. The final step is to use the mathematical solution to make predictions about the future behaviour of the original system. Correspondingly, a patient's symptoms have to be sorted out in order that the doctor can make a diagnosis and decide what he thinks the disease is that needs treatment, which he then prescribes. The final state again is that of forward-looking or prognosis. The validation stage in the modelling procedure is vital. The results of the model must be comnared with as many of the known facts as possible in order to give reasonable confidence in the prediction of new facts. Mathematical modelling in medicine is both a potentially useful tool for practising surgeons as well as beinz a challenge to applied mathematicians. Clearly, collaboration between the various specialists will pro(luce the most promising
Medical men are accustomed to the use of physical models in their work in order to illustrate various organs of the body, what they look like, and how they function and to study results. how to control them. A physical model is only a partial representation of the real thing. It will always be imperfect in some sefise; otherwvise it would be too complicated to be useful. A mathematical model, similarly, is a partial description or representation in mathematical terms of a real situation, but the building bricks in this case are the usual symbols of mathematics-for example, x, y, +, -,- -put together to form equations. These equations constitute the mathematical model, and the behaviour of quantities in the equations simulates the behaviour of the corresponding physical entity in the real problem. We have to decide what featuires in the real problem we are going to concentrate on in our model and reject those we feel to be unim-
FIG. 4
Mathematics and the surgeon
303
The generality of mathematics A characteristic feature of mathematics is its generality or commonality, by which we mean that a particular mathematical model often describes widely different situations in real life. Sometimes this is fairly obvious and accepted almost unconsciously. Although the man in the street accepts that the process of adding 2 oranges to 2 more oranges to give 4 is the same as adding 2 legs to 2 legs to give 4 legs, this is, in fact, a powerful process of abstraction which children have to, be taught. The well-known example taken from a school textbook, 'King Solomon had 122 wives and 243 concubines. Add the two together and state the result', shows that simple arithmetic is not always enough. The important stibject of statistics is a second illustration of the generality of mathematical technioues-for example, the evaluation of the mean andi the standard deviation of the heights of boys between the ages of 5 and I5 is mathematically identical with the corresponding calculation applied to a blood count. The subject of operational research provides mathematical techniques for planning and decision-making which are applicable in many different kinds of situation-for example, the erection of a complicated building; the efficient control of an ambulance service; the provision of a suitable diet at minimum cost. There is, however, a deeper level of generalitv which emerges only when the mathematical equations themselves are inspected. Thus it is not obvious at first sight that the same mathematical model can be used to describe heat flow through the wall of a building, diffusion of an anaesthetic in the body, and the distribution of electric potential. Diffusion is a general mechanism by which oxygen, nutrients, waste products, drugs, etc, are distributed through the body.
presses a wealth of information into a usable form in which a user can absorb and assess the information. A simple example is the formula relating the volume of a sphere, V, to its radius, r-that is, V = 4/3 (7Tr3). From it we can easily discover what is the volume of any sphere if we know its radius. Sometimes instead of formulae we have tables of values or graphs, and nowadays information is kept in a computer and can be retrieved and manipulated at will. Not all information is known with certainty and precision, however. Sometimes we can only quote the chance or probability of something happening. To purify and interpret such data we need the subject of statistics. Medical facts are frequently of the kind that call for a statistical interpretation. Questions arise such
Different uses of a mathematical model i) To provide a compact and convenient quantitative description of observed data A mathematical expression which summarizes results may be empirical only, since it has no knowvn theoretical foundation, or it may have been derived by a chain of logical arguments starting from widely accepted postulates or axioms. In either case the expression com-
2) To provide a logical framework for planning and decision-making The subject known as 'operational research' means the use of mathematical, statistical, and computer techniques to help the planning of cornDlicated tindertakings and projects and to do it in the most efficient way. 'Efficiency' needs to be defined in each case. It could mean at minimum cost or with maximum yield of a
as :
a) Can we use a single value for example, a mean value-to represent comethina about a collection of data? If we do, how and to what extent are the crude observations spread about the mean? b) How can we define and measure the relation between two sets of variables and what reliability can we put on our measurement? This is the subject of correlation. c) When a property has to be judged by extracting a sample from a large collection or population, how large should the sample be to justify certain conclusions about the population and what reliability may wve attach to our conclusions? d) Is a measured difference real or fortuitous?
How significant are the differences in
weights of two groups of children fed on two different test diets? It is in the nature of much of the data with which doctors must deal that mathematical models of a statistical kind are of overwhelming importance in medicine.
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particular product or as quickly as possible. There are different branches of operational research-for example: a) Critical path analysis, which shows how to economise the overall time to completion of a project which involves many hundreds, perhaps, of activities that must follow each other in a certain logical order-for example, the building of a hospital. b) Transportation, which obviously means how to transport supplies of one sort or another from a number of depots to a number of demand points, customers, or patients in such a way as to minimize the total cost of the exercise. c) The allocation of limited resources or the blending of constituent ingredients, in each the objective being to satisfy certain constraints at minimum cost or maximum profit. etc. One example would be how to spend a capital grant to purchase several types of a particular machine; some types are more efficient than others in, may, treating patients, btut cost more. How many of each type should be bought to satisfy the needs of as many patients as possible? Another could be that a number of ambulances are needed simultaneously at several road accidents. The ambulances are available at a number of local ambulance stations; how should they be routed to the different accident sites in order to minimize the cost of the whole operation? The branch of mathematics nowadays referred to as decision theory is essentially applied pirobability theory. The prevention of deep vein thrombosis following mild cardiac infarction described by Dr Emerson4 is a simple example of the place of decision theorv in deciding whether and when patients should be given prophylactic heparin therapy.
3) The prediction of the behaviour of a
One of the major reasons for the emplovment of mathematicians and the u^-e of large computers by industry is the hope that together they may be able to predict howv some chemical plant or some engineering structure is likely to behave under different conditions. Ideally, such theoretical predictions are expected to be quickel, less expensive, ,'nd more convenient than the alternative, which usiially system
means experimenting with the real system by trial-and-error methods. One fruitful and popular field at the present time for the use of mathematical models is what the engineers call 'structural analysis'. The structure may be a roof span, a pillar, a cathedral arch, an aircraft fuselage, a dam. All are load-bearing structures with definite elastic properties and in some cases both their static and dynamic responses to externally applied loads may be important. A fairly new branch of mathematics called 'finite element analysis' was developed first by engineers5 and is now widely used by engineers and mathematicians alike' in structural analysis. These same methods are currently being applied to study the mechanical stresses and strains in various parts of the human body. Perhaps I may be allowed to refer to work at my own university without wishing to give the impression that the efforts of other similar groups are unimportant. The work in the Department of Mechanical Engineering at Brunel University is being carried out by Mr A Yettram and his students. Some of the work on teeth in collaboration with Professor H M Pickard, of the Royal Dental Hospital, has already been reported7. Other work is proceeding with Professor Scales, of the Royal National Orthopaedic Hospital. Stanmore, on problems associated with a hip prosthesis. Mr C Vinson is analysing data from the Brompton Hospital in a study of the elastic properties of the human heart. In the mathematical model of a hip prosthesis the metal insert is divided into a number of small elements joined together at a discrete number of nodes. A mathematical approximation to the strain within each element is postulated and a set of linear algebraic equations is drawn up which express the equilibrium conditions of the stresses acting at each node. The computerized solution of this large system of equations yields the displacement of each node under a given external load and hence the distribution of stresses within the metal insert. In particular, regions of high stress can be identified and, it is hoped, the metal insert can be modified to produce a more acceptable distribution of stress. Finite-element stress analyses of human cardiac structures are reported! for exaniple,
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by Hamid and Ghistas. They study the stress tion to be obtained from surface measurements distribution in the left ventricular chamber of about the behaviour inside a tumour. the heart; their mathematical model incorporates the true shape of the actual chamber. 5) Relating non-invasive measurements Clearly it is They can study, for example, the stresses to internal behaviour around an infarcted area. The same paper re- much easier to make measurements at the ports a stress analysis of the aortic valve in mouth of a patient than in his alveoli. Attempts order to help prescribe the variable thickness are being made" to model the lungs mathemaof a prosthetic valvular leaflet. tically with the ultimate intention of using the possible and convenient measurements to give us information about what is happening deep 4) The interpretation of measurements in the lung system. It would be particularly Mathematics is frequently the essential link helpful if the values of parameters in the between the measured data and the primary model could be correlated with mathematical structural information sought. The reconstrucconditions of the lungs. various diseased tion of objects in three dimensions from their projections-for example, electron micrographs 6) The quantitative evaluation of theories or X-ray pictures-is made possible only by All science proceeds of mechanism whether analysis, the underlying mathematical this be the so-called algebraic reconstruction by postulating mechanisms and theories to techniques in which a very large set of account for known facts and then testing the algebraic equations has to be solved by a hypotheses by carrying out further expericomputer by the application of Fourier trans- ments. Any quantitative hypothesis is clothed form methods or by the more recent use of in mathematical terms. One of the great adrecursive filtering techniques. The brain-scan- vantages is that any factor can be changed in ning and body-scanning machines developed isolation in a mathematical model, whereas bv EMI, for example, incorporate mathema- in real life it is often difficult not to change two or three factors simultaneously. Furthertics of this nature. Another example can be taken from medical more, a mathematical model of part of a research into the absorption of oxygen by the patient's anatomy can be manipulated without cells of a tumour. The killing power of high- any shock to the patient's system-for exor energy radiation depends among other factors ample, we can cut off his intake of oxygen double his pulse rate to suit the mathemation the oxygen content of the irradiated cells. This varies from one part of a tumour to an- cian's whim. A recent article in the Journal other, but it is not easy to observe the variation of Theoretical Biology"2, for example, describes a mathematical model for exploring the sugdirectly. A mathematical model can be constructed gestion that the blood flow patterns in the to represent the process of simultaneous dif- neighbourhood of bifurcations in the branching fusion and absorption of oxygen within a layer network of tubes forming the cardiovascular of tumorous tissue9"0. From the model the system tend to produce lesions. time rate of change of oxygen concentration in different parts of an exposed surface of the Collaboration laver can be calculated. By comparing these It is clear that successful mathematical modelcalculated rates with those measured by an ling in medicine depends on a high degree of collaboration between specialists in various array of metal electrodes located in the sealed surface the rate of oxygen consumption at areas. We are led at once to think of teams internal points of the tumour beneath each of workers which will necessarily include a electrode, which apnears as an adiustable para- mathematical consultant. The Institute of meter in the equiations, can be dediuced. The Mathematics and its Applications is keen to next dosare of radiation can then be directed promote collaborative ventures. To this end at the cells of high oxypen content to secure there is in existence a Subcommittee on Mathemaximum efficiency of killino. Tn this example matics, Medicine, and Biology on which sit also, therefore, mathematics enables informa- a number of members of the medical profession
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as well as mathematicians. The Scottish Council for Postgraduate Medical Education alsol has a Joint Working Party on Mathematics in the Practice of Medicine Several interdisciplinary conferences have already been organized and others are planned. They include: mathematics in clinical medicine; medical and biological applications of statistics; mathematics of biological rhythms; mathematics in the human sciences; diffusion in medicine and biology; and the teaching of statistics to medical, biological, and agricultural students. That these conferences are attracting good support is indicative of the widespread interest in mathematical models of medical problems.
Postgraduate and postexperience studies Accepting that much of the research work in medicine is likely to be most successful and productive if carried otut by interdisciplinary teams and that it is desirable that at least one member of the team shall have specialist mathematical knowledge, the question arises as to how best this member should be trained. Should he be a graduate mathematician who has afterwards acquired a knowledge of the relevant parts of medicine or should he be basically a medical man who has undergone a course of postgraduate training in mathematics? Whatever the merits of the two routes, no doubt we shall need to make full use of both of them to maintain an adequate volume of medical research in which mathematics has its proper supporting role. Our concern in this paper must be with the mathematical conversion of a medical graduate. How can we decide what parts of mathematics to present to him and how should the approach be made? The second point may well be more important than the first. Above all else we must find ways of exciting his interest in mathematics, probably by demonstrating its relevance for him via mathematical models. His interest will almost always lie in apnlying mathematical techniques to proh lems in his own field rather than in rigorous nroofs, elegant thouch they may seem to professional mathematicians. It is incumbent upon the mathematics lecturer or tutor to learn the medical 1anpruape of his punil and show an interest in his problems.
As to content, no doubt everyone would wish to include some statistics and probability theory, in some cases far more than an elementary introduction. As to mathematics proper, no doubt if one were to read widely enough in a representative selection of medical journals one could find an example of almost every branch of mathematics in use somewhere. Nevertheless, some branches of mathematics are likely to be more rewarding than others for the mathematics specialist in a medical team. One way of drawing up a syllabus is to list the constituent parts of the human body and their physiological functions. We must also incltude the instruments and techniques used in the analysis and treatment of patients' ills and in the underlving research. This would be followed by a second list of the fundamental physical, chemical, biological, and enzineering processes involved. This in turn would define the mathematical topics most likely to be involved. The accompanying table is not intended to be comnrehensive but to illustrate these general remarks. A full analysis would need to be undertaken by a multidisciplinarian team. Certain conclusions may be reached concerning the mathematical content of such a
syllabus: a) We have already agreed that statistics is to be included in the list of topics. b) Some acquaintance with computers, their programming and organization, is essential. The depth of instruction needed will vary a great deal. c) An introduction to the relevant parts of numerical analysis is also essential if a medical research worker is to understand how a computer deals with a large set of algebraic equations or a partial differential equation, for example. At least he can then know when to approach a mathematical consultant or may wish to proceed to write his own computer programs. Tt is not widely recognized that the computer's approach to a mathematical solution is freauently quite different from the classical analytical methods taught in undergraduate courses. Computers offer the opnortunity of a new start with numerical methods in which the operations are those of simple arithmetic"3.
Mathematics and the surgeon Human body Skeleton and muscles
Physical description
Load-bearing structure
307
Mathematics involved
Mechanics; elasticity; matrix algebra;
ordinary differential equations and partial differential equations; finite differences; finite elements
Cardiovascular system
Circulation of blood through a branched system
Fluid flow along tubes; partial differential equations
Pulmonary airways
Gas flow and diffusion
Partial differential cquations; graph theory
Brain and nervous system
A system of interconnected nerve cells which transmits information as electrical impulses
Information and control theory; cybemetics; electrical circuit theory;
Digestive system
Diffusion and chemical reactions
Differential equations
Diseases
Population growth of viruses
Exponential growth and decay;
and bacteria; natural and artificial control
d) There will be a need for one or more intermediate courses dealing with the basic mathematical concepts and skills on which the topics listed in the table rest. Calculus, exponential functions, matrix algebra, functions of more than one variable, and numerical integration are obvious examples.
Medical--mathematical co-operation The world as a whole and, I imagine, the
logic; topology
differential equations
has to wait for suitable advances in mathematics itself, just as developments in practical surgery can be held up till new practical skills or instruments become available. What seems to be most needed at the present time is to extend the areas of collaboration between the professions of mathematics and medicine. We need more mathematicians who are aware of medical problems and who are willing and able to try to build mathematical models of them. We need more doctors, including surgeons, who know enough about mathematics at least to be able to describe to a mathematician potentially fruitful problems. The key words are 'collaboration' and 'teamwork'.
medical profession can be divided into two groups-those who consider all mathematicians to be totally useless people who are best left to work out of harm's way in their ivory towers and, at the other extreme, those who regard a mathematician as all-powerful. They feel that with his kitbag of mathematical techniques and a powerful computer he could, if he so wished, solve any practical problem References I Parkhouse, J (I974) Bulletin of the Institute of put to him. The 'place of mathematics' in Mathematics and its Applications, Io, 6o. various professions has long been hotly debated 2 Cruickshank, E. K (I974) Bulletin of the Instiand a current topic about which widely divertute of Mathematics and its Applications, Io, 58. Aitchison, J (I974) Bulletin of the Institute of gent views are held is that of computer-aided Mathenatics and its Applications, io, 48. medical diagnosis. M Emerson, P A (I974) Bulletin of the Institute The truth is, of course, that there are some of Mathematics and its Applications, IO, 33. situations in which the mathematician or 5 Zinckiewicz, 0 C (I97I) Finite Element Method in Engineering Science. New York, McGraw statistician can be of great assistance. Equally, there are many medical areas for which no 6 Hill WVhiteman, J (ed.) (I973) The Mathematics of really satisfactory mathematical techniques Finite Elements and Applications. New York and exist as yet. The solution of problems often London, Academic Press. -
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7 Wright, K W J (1975) PhD Thesis, Brunel University. 8 Hamid, M S, aind Ghista, D N (I974) Proceedings of the I974 International Conference on Finite Element Methods in Engineering held at the University of New South WVales, Australia (ed. Pulmano, V A, and Kabaila, A P), P. 337. Australia, Clarendon Press. 9 Evans, N T S, and Gourlay, A R journal of the Institute of Mathematics and its Applications. In press.
io Crank, J, and Gupta, R S (1972) Journal of the Institute of Mathematics and its Applications, I0, 19, 296. ii Pack, A I, Murray-Smith, D, Mills, R J, Hooper, M, and Taylor, J (I974) Bulletin of the Institute of Mathematics and its Applications, IO, 20 12 Zamir, M, and Roach, M R (I973) Journal of Theoretical Biology, 42, 33. 13 Crank, J (I974) Bulletin of the Institute of Mathematics and its Applications, 1o, 42.
Mathematics and the J
Crank
DSC
FIIlStP
surgeon
FIMA
Professor and Head of the School of Mathematical Studies, Brunel University, Uxbridge, Middlesex
Summary The surgeon uses elementary mathematics just as much as any other educated layman. In his professional life, however, much of the knowledge and skill on which he relies has had a mathematical strand in its development, possibly woven into the supporting disciplines such as physics, chemistry, biology, and bioengineering. The values and limitations of mathematical models are examined briefly in the general medical field and particularly in relation to the surgeon. Arithmetic and statistics are usually regarded as the most immediately useful parts of mathematics. Examples are cited, however, of medical postgraduate work which uses other highly advanced mathematical techniques. The place of mathematics in postgraduate and postexperience teaching courses is touched on. The role of a mathematical consultant in the medical team is discussed. Introduction It could be argued that Brunel University owes its name to a igth-century application of mathematics in surgery. In I 843 I K Brunel, the famous engineer, swallowed a half-sovereign which lodged in his windpipe. A tracheotomy operation performed by the eminent surgeon Sir Benjamin Brodie proved unsuccessful. Brunel himself then designed a simple apparatus to which he was strapped and spun rapidly head over heels. At the second attempt, under the action of the so-called centrifugal force, the coin was dislodged and dropped from his mouth. Brunel lived and a university now bears his name. It seems natural to start by commenting on the title to which the organizers of the symposium invited me to speak. Although nerhans lacking in literary comnatibility, it does indicate precisely the relationshin we wish to consider. One alternative. 'The mathematician and ihe surgeon', would have concentrated
attention on a more personal relationship than is intended and would have implied that all the mathematics of value to the surgeon comes to him through mathematicians, which is certainly not true. On the other hand, 'Mathematics and surgery' would overlook the fact that a surgeon's use of mathematics is not confined to the operating theatre and would suggest an academic discussion of the intersecting areas of the two specialisms. First of all, a surgeon is an intelligent layman and as such must be able to count, perform elementary arithmetic, and even understand simple graphs and statistics. These basic skills and concepts are expected of every educated person. To quote Professor Parkhousel, 'This is comparable to the fact that we, as doctors, would expect any mathematician to have a general knowledge of hygiene, first-aid, child care and development, nutrition, emergency resuscitation, and so forth'. It is difficult to make a case for more than simple mathematical techniques for the man in the street. As a graduate in medicine, however, the surgeon will have studied more mathematics, usually included in physics and chemistry, biology, biochemistry, and perhaps even anatomy and bioengineering. A practising surgeon is caught up in a web of mathematics, for the most part invisible and of which it could be argued, and is, that he need know nothing. There are obvious parallels with people who drive cars without understanding anything aboiut the internal combustion engine or cheerfully watch television without understanding the electronic circuitry. If I may quote Professor Cruickshank2, 'One is inclined to forget that as recently as one huindred years ago empiricism dominated the practice of the "art" of medicine, and "science" was just beginning to influence the thinking of the medical practitioner. It is the ranid development of physics and chemistry, under-
Delivered at the Annual Meeting of Fellows and Members on ioth December I975 as part of a symposium on 'Teaching postgraduate surgery'
Mathematics and the surgeon
D-IM
301
1144
¶RfE AL \46 e L I
c c
L D
I
s
MArtHl"
sslt~~~mm t tWuoOc -
Woiw3-
3o
I G1
.
'I I e,
A/J
FIG. I
pinned by mathematics, that has made all our advances possible: the development of the microscope; the discovery and use of radiation for diagnosis and therapy; the explosion of clinical laboratory investigations (doubling in number every five years) for diagnosis and prognosis; and the enormous expansion of drug therapy, where io-6 mg of a substance can radically alter cellular function and reproduction-all these virtually within the lifetime of most of us-are testimony to what we owe to the application of mathematics to the physical and biological sciences'. At every surgeon's elbow is not only a nurse, but a mathematician. It is at postgraduate level, however, whether taught courses, seminars, conferences, or research, that the surgeon is most likely to find a need to extend his knowledge of mathematics. We shall return to a consideration of postgraduate work later, but sufficient to say here that even a casual reading of medical research journals reveals how much of the work depends on mathematical techniques. Mathematical models Before we look at the special interest mathematics may have for the siurgeon it is worth while to examine a little more carefully the process of applying mathematics to any practical problem of real life. Somehow we have to express
our real problem in such a way that mathematical techniques which are essentially abstract can be brought to bear. The process has been described by Professor Synge in the following way: 'The use of applied mathematics in its relation to a physical problem involves three stages; a dive from the world of reality into the world of mathematics; a swim in the world of mathematics; a climb from the world of mathematics into the world of reality, carrying a prediction in our teeth'. Figure gives an artist's impression of this diving and swimi
ming.
In the mathematical world we use mathematical language. Both our professions are guilty of using specialist language or jargon and forgetting that this is probably not understood by those outside the particular 'field. Professor Aitchison3 has identified a 'newmath syndrome' with the following symptoms: Algebritis-an obsessive fear of operations, particularly of the algebraic type. Binary paralexia-an inability to manipulate and coordinate mathematical symbols. Calculus deficiency-a failure to integrate properly. They could easily be accepted as genuine diseases by an innocent reader. Mathematicians use diagrams, graphs, and dy . As an example, a symbols-x, y, 0, dx
302
J Crank
FIG. 2
problem referring to a seal sitting on an ice floe could well occur in school mathematics, leading to the question, 'If the ice floe tips up, how long does it take for the seal to enter the water?' Any sixth-fonrmer would draw a diagram and one might expect Figure 2, but he will instead draw Figure 3 and expect everybody to understand that this represents a seal on an ice floe. In fact, he has unconsciously performed a process that we call abstraction, or the building of a mathematical model.
FIG. 3
portant. We make assumptions sufficiently simple to reduce the problem to one that we can handle mathematically but not so simple that our results will be meaningless. Whether a particular assumption is acceptable or not depends on the use to which a given model is to be put. So mathematical modelling is a skilful art. I venture to suggest that there is a close parallel between this modelling and the medical treatment of a patient. Figure 4 illustrates this analogy. In the mathematical case we proceed from the real problem, having decided what are the really vital features, to the model and thence to a suitable mathematical treatment of our equations. The final step is to use the mathematical solution to make predictions about the future behaviour of the original system. Correspondingly, a patient's symptoms have to be sorted out in order that the doctor can make a diagnosis and decide what he thinks the disease is that needs treatment, which he then prescribes. The final state again is that of forward-looking or prognosis. The validation stage in the modelling procedure is vital. The results of the model must be comnared with as many of the known facts as possible in order to give reasonable confidence in the prediction of new facts. Mathematical modelling in medicine is both a potentially useful tool for practising surgeons as well as beinz a challenge to applied mathematicians. Clearly, collaboration between the various specialists will pro(luce the most promising
Medical men are accustomed to the use of physical models in their work in order to illustrate various organs of the body, what they look like, and how they function and to study results. how to control them. A physical model is only a partial representation of the real thing. It will always be imperfect in some sefise; otherwvise it would be too complicated to be useful. A mathematical model, similarly, is a partial description or representation in mathematical terms of a real situation, but the building bricks in this case are the usual symbols of mathematics-for example, x, y, +, -,- -put together to form equations. These equations constitute the mathematical model, and the behaviour of quantities in the equations simulates the behaviour of the corresponding physical entity in the real problem. We have to decide what featuires in the real problem we are going to concentrate on in our model and reject those we feel to be unim-
FIG. 4
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The generality of mathematics A characteristic feature of mathematics is its generality or commonality, by which we mean that a particular mathematical model often describes widely different situations in real life. Sometimes this is fairly obvious and accepted almost unconsciously. Although the man in the street accepts that the process of adding 2 oranges to 2 more oranges to give 4 is the same as adding 2 legs to 2 legs to give 4 legs, this is, in fact, a powerful process of abstraction which children have to, be taught. The well-known example taken from a school textbook, 'King Solomon had 122 wives and 243 concubines. Add the two together and state the result', shows that simple arithmetic is not always enough. The important stibject of statistics is a second illustration of the generality of mathematical technioues-for example, the evaluation of the mean andi the standard deviation of the heights of boys between the ages of 5 and I5 is mathematically identical with the corresponding calculation applied to a blood count. The subject of operational research provides mathematical techniques for planning and decision-making which are applicable in many different kinds of situation-for example, the erection of a complicated building; the efficient control of an ambulance service; the provision of a suitable diet at minimum cost. There is, however, a deeper level of generalitv which emerges only when the mathematical equations themselves are inspected. Thus it is not obvious at first sight that the same mathematical model can be used to describe heat flow through the wall of a building, diffusion of an anaesthetic in the body, and the distribution of electric potential. Diffusion is a general mechanism by which oxygen, nutrients, waste products, drugs, etc, are distributed through the body.
presses a wealth of information into a usable form in which a user can absorb and assess the information. A simple example is the formula relating the volume of a sphere, V, to its radius, r-that is, V = 4/3 (7Tr3). From it we can easily discover what is the volume of any sphere if we know its radius. Sometimes instead of formulae we have tables of values or graphs, and nowadays information is kept in a computer and can be retrieved and manipulated at will. Not all information is known with certainty and precision, however. Sometimes we can only quote the chance or probability of something happening. To purify and interpret such data we need the subject of statistics. Medical facts are frequently of the kind that call for a statistical interpretation. Questions arise such
Different uses of a mathematical model i) To provide a compact and convenient quantitative description of observed data A mathematical expression which summarizes results may be empirical only, since it has no knowvn theoretical foundation, or it may have been derived by a chain of logical arguments starting from widely accepted postulates or axioms. In either case the expression com-
2) To provide a logical framework for planning and decision-making The subject known as 'operational research' means the use of mathematical, statistical, and computer techniques to help the planning of cornDlicated tindertakings and projects and to do it in the most efficient way. 'Efficiency' needs to be defined in each case. It could mean at minimum cost or with maximum yield of a
as :
a) Can we use a single value for example, a mean value-to represent comethina about a collection of data? If we do, how and to what extent are the crude observations spread about the mean? b) How can we define and measure the relation between two sets of variables and what reliability can we put on our measurement? This is the subject of correlation. c) When a property has to be judged by extracting a sample from a large collection or population, how large should the sample be to justify certain conclusions about the population and what reliability may wve attach to our conclusions? d) Is a measured difference real or fortuitous?
How significant are the differences in
weights of two groups of children fed on two different test diets? It is in the nature of much of the data with which doctors must deal that mathematical models of a statistical kind are of overwhelming importance in medicine.
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particular product or as quickly as possible. There are different branches of operational research-for example: a) Critical path analysis, which shows how to economise the overall time to completion of a project which involves many hundreds, perhaps, of activities that must follow each other in a certain logical order-for example, the building of a hospital. b) Transportation, which obviously means how to transport supplies of one sort or another from a number of depots to a number of demand points, customers, or patients in such a way as to minimize the total cost of the exercise. c) The allocation of limited resources or the blending of constituent ingredients, in each the objective being to satisfy certain constraints at minimum cost or maximum profit. etc. One example would be how to spend a capital grant to purchase several types of a particular machine; some types are more efficient than others in, may, treating patients, btut cost more. How many of each type should be bought to satisfy the needs of as many patients as possible? Another could be that a number of ambulances are needed simultaneously at several road accidents. The ambulances are available at a number of local ambulance stations; how should they be routed to the different accident sites in order to minimize the cost of the whole operation? The branch of mathematics nowadays referred to as decision theory is essentially applied pirobability theory. The prevention of deep vein thrombosis following mild cardiac infarction described by Dr Emerson4 is a simple example of the place of decision theorv in deciding whether and when patients should be given prophylactic heparin therapy.
3) The prediction of the behaviour of a
One of the major reasons for the emplovment of mathematicians and the u^-e of large computers by industry is the hope that together they may be able to predict howv some chemical plant or some engineering structure is likely to behave under different conditions. Ideally, such theoretical predictions are expected to be quickel, less expensive, ,'nd more convenient than the alternative, which usiially system
means experimenting with the real system by trial-and-error methods. One fruitful and popular field at the present time for the use of mathematical models is what the engineers call 'structural analysis'. The structure may be a roof span, a pillar, a cathedral arch, an aircraft fuselage, a dam. All are load-bearing structures with definite elastic properties and in some cases both their static and dynamic responses to externally applied loads may be important. A fairly new branch of mathematics called 'finite element analysis' was developed first by engineers5 and is now widely used by engineers and mathematicians alike' in structural analysis. These same methods are currently being applied to study the mechanical stresses and strains in various parts of the human body. Perhaps I may be allowed to refer to work at my own university without wishing to give the impression that the efforts of other similar groups are unimportant. The work in the Department of Mechanical Engineering at Brunel University is being carried out by Mr A Yettram and his students. Some of the work on teeth in collaboration with Professor H M Pickard, of the Royal Dental Hospital, has already been reported7. Other work is proceeding with Professor Scales, of the Royal National Orthopaedic Hospital. Stanmore, on problems associated with a hip prosthesis. Mr C Vinson is analysing data from the Brompton Hospital in a study of the elastic properties of the human heart. In the mathematical model of a hip prosthesis the metal insert is divided into a number of small elements joined together at a discrete number of nodes. A mathematical approximation to the strain within each element is postulated and a set of linear algebraic equations is drawn up which express the equilibrium conditions of the stresses acting at each node. The computerized solution of this large system of equations yields the displacement of each node under a given external load and hence the distribution of stresses within the metal insert. In particular, regions of high stress can be identified and, it is hoped, the metal insert can be modified to produce a more acceptable distribution of stress. Finite-element stress analyses of human cardiac structures are reported! for exaniple,
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by Hamid and Ghistas. They study the stress tion to be obtained from surface measurements distribution in the left ventricular chamber of about the behaviour inside a tumour. the heart; their mathematical model incorporates the true shape of the actual chamber. 5) Relating non-invasive measurements Clearly it is They can study, for example, the stresses to internal behaviour around an infarcted area. The same paper re- much easier to make measurements at the ports a stress analysis of the aortic valve in mouth of a patient than in his alveoli. Attempts order to help prescribe the variable thickness are being made" to model the lungs mathemaof a prosthetic valvular leaflet. tically with the ultimate intention of using the possible and convenient measurements to give us information about what is happening deep 4) The interpretation of measurements in the lung system. It would be particularly Mathematics is frequently the essential link helpful if the values of parameters in the between the measured data and the primary model could be correlated with mathematical structural information sought. The reconstrucconditions of the lungs. various diseased tion of objects in three dimensions from their projections-for example, electron micrographs 6) The quantitative evaluation of theories or X-ray pictures-is made possible only by All science proceeds of mechanism whether analysis, the underlying mathematical this be the so-called algebraic reconstruction by postulating mechanisms and theories to techniques in which a very large set of account for known facts and then testing the algebraic equations has to be solved by a hypotheses by carrying out further expericomputer by the application of Fourier trans- ments. Any quantitative hypothesis is clothed form methods or by the more recent use of in mathematical terms. One of the great adrecursive filtering techniques. The brain-scan- vantages is that any factor can be changed in ning and body-scanning machines developed isolation in a mathematical model, whereas bv EMI, for example, incorporate mathema- in real life it is often difficult not to change two or three factors simultaneously. Furthertics of this nature. Another example can be taken from medical more, a mathematical model of part of a research into the absorption of oxygen by the patient's anatomy can be manipulated without cells of a tumour. The killing power of high- any shock to the patient's system-for exor energy radiation depends among other factors ample, we can cut off his intake of oxygen double his pulse rate to suit the mathemation the oxygen content of the irradiated cells. This varies from one part of a tumour to an- cian's whim. A recent article in the Journal other, but it is not easy to observe the variation of Theoretical Biology"2, for example, describes a mathematical model for exploring the sugdirectly. A mathematical model can be constructed gestion that the blood flow patterns in the to represent the process of simultaneous dif- neighbourhood of bifurcations in the branching fusion and absorption of oxygen within a layer network of tubes forming the cardiovascular of tumorous tissue9"0. From the model the system tend to produce lesions. time rate of change of oxygen concentration in different parts of an exposed surface of the Collaboration laver can be calculated. By comparing these It is clear that successful mathematical modelcalculated rates with those measured by an ling in medicine depends on a high degree of collaboration between specialists in various array of metal electrodes located in the sealed surface the rate of oxygen consumption at areas. We are led at once to think of teams internal points of the tumour beneath each of workers which will necessarily include a electrode, which apnears as an adiustable para- mathematical consultant. The Institute of meter in the equiations, can be dediuced. The Mathematics and its Applications is keen to next dosare of radiation can then be directed promote collaborative ventures. To this end at the cells of high oxypen content to secure there is in existence a Subcommittee on Mathemaximum efficiency of killino. Tn this example matics, Medicine, and Biology on which sit also, therefore, mathematics enables informa- a number of members of the medical profession
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as well as mathematicians. The Scottish Council for Postgraduate Medical Education alsol has a Joint Working Party on Mathematics in the Practice of Medicine Several interdisciplinary conferences have already been organized and others are planned. They include: mathematics in clinical medicine; medical and biological applications of statistics; mathematics of biological rhythms; mathematics in the human sciences; diffusion in medicine and biology; and the teaching of statistics to medical, biological, and agricultural students. That these conferences are attracting good support is indicative of the widespread interest in mathematical models of medical problems.
Postgraduate and postexperience studies Accepting that much of the research work in medicine is likely to be most successful and productive if carried otut by interdisciplinary teams and that it is desirable that at least one member of the team shall have specialist mathematical knowledge, the question arises as to how best this member should be trained. Should he be a graduate mathematician who has afterwards acquired a knowledge of the relevant parts of medicine or should he be basically a medical man who has undergone a course of postgraduate training in mathematics? Whatever the merits of the two routes, no doubt we shall need to make full use of both of them to maintain an adequate volume of medical research in which mathematics has its proper supporting role. Our concern in this paper must be with the mathematical conversion of a medical graduate. How can we decide what parts of mathematics to present to him and how should the approach be made? The second point may well be more important than the first. Above all else we must find ways of exciting his interest in mathematics, probably by demonstrating its relevance for him via mathematical models. His interest will almost always lie in apnlying mathematical techniques to proh lems in his own field rather than in rigorous nroofs, elegant thouch they may seem to professional mathematicians. It is incumbent upon the mathematics lecturer or tutor to learn the medical 1anpruape of his punil and show an interest in his problems.
As to content, no doubt everyone would wish to include some statistics and probability theory, in some cases far more than an elementary introduction. As to mathematics proper, no doubt if one were to read widely enough in a representative selection of medical journals one could find an example of almost every branch of mathematics in use somewhere. Nevertheless, some branches of mathematics are likely to be more rewarding than others for the mathematics specialist in a medical team. One way of drawing up a syllabus is to list the constituent parts of the human body and their physiological functions. We must also incltude the instruments and techniques used in the analysis and treatment of patients' ills and in the underlving research. This would be followed by a second list of the fundamental physical, chemical, biological, and enzineering processes involved. This in turn would define the mathematical topics most likely to be involved. The accompanying table is not intended to be comnrehensive but to illustrate these general remarks. A full analysis would need to be undertaken by a multidisciplinarian team. Certain conclusions may be reached concerning the mathematical content of such a
syllabus: a) We have already agreed that statistics is to be included in the list of topics. b) Some acquaintance with computers, their programming and organization, is essential. The depth of instruction needed will vary a great deal. c) An introduction to the relevant parts of numerical analysis is also essential if a medical research worker is to understand how a computer deals with a large set of algebraic equations or a partial differential equation, for example. At least he can then know when to approach a mathematical consultant or may wish to proceed to write his own computer programs. Tt is not widely recognized that the computer's approach to a mathematical solution is freauently quite different from the classical analytical methods taught in undergraduate courses. Computers offer the opnortunity of a new start with numerical methods in which the operations are those of simple arithmetic"3.
Mathematics and the surgeon Human body Skeleton and muscles
Physical description
Load-bearing structure
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Mathematics involved
Mechanics; elasticity; matrix algebra;
ordinary differential equations and partial differential equations; finite differences; finite elements
Cardiovascular system
Circulation of blood through a branched system
Fluid flow along tubes; partial differential equations
Pulmonary airways
Gas flow and diffusion
Partial differential cquations; graph theory
Brain and nervous system
A system of interconnected nerve cells which transmits information as electrical impulses
Information and control theory; cybemetics; electrical circuit theory;
Digestive system
Diffusion and chemical reactions
Differential equations
Diseases
Population growth of viruses
Exponential growth and decay;
and bacteria; natural and artificial control
d) There will be a need for one or more intermediate courses dealing with the basic mathematical concepts and skills on which the topics listed in the table rest. Calculus, exponential functions, matrix algebra, functions of more than one variable, and numerical integration are obvious examples.
Medical--mathematical co-operation The world as a whole and, I imagine, the
logic; topology
differential equations
has to wait for suitable advances in mathematics itself, just as developments in practical surgery can be held up till new practical skills or instruments become available. What seems to be most needed at the present time is to extend the areas of collaboration between the professions of mathematics and medicine. We need more mathematicians who are aware of medical problems and who are willing and able to try to build mathematical models of them. We need more doctors, including surgeons, who know enough about mathematics at least to be able to describe to a mathematician potentially fruitful problems. The key words are 'collaboration' and 'teamwork'.
medical profession can be divided into two groups-those who consider all mathematicians to be totally useless people who are best left to work out of harm's way in their ivory towers and, at the other extreme, those who regard a mathematician as all-powerful. They feel that with his kitbag of mathematical techniques and a powerful computer he could, if he so wished, solve any practical problem References I Parkhouse, J (I974) Bulletin of the Institute of put to him. The 'place of mathematics' in Mathematics and its Applications, Io, 6o. various professions has long been hotly debated 2 Cruickshank, E. K (I974) Bulletin of the Instiand a current topic about which widely divertute of Mathematics and its Applications, Io, 58. Aitchison, J (I974) Bulletin of the Institute of gent views are held is that of computer-aided Mathenatics and its Applications, io, 48. medical diagnosis. M Emerson, P A (I974) Bulletin of the Institute The truth is, of course, that there are some of Mathematics and its Applications, IO, 33. situations in which the mathematician or 5 Zinckiewicz, 0 C (I97I) Finite Element Method in Engineering Science. New York, McGraw statistician can be of great assistance. Equally, there are many medical areas for which no 6 Hill WVhiteman, J (ed.) (I973) The Mathematics of really satisfactory mathematical techniques Finite Elements and Applications. New York and exist as yet. The solution of problems often London, Academic Press. -
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7 Wright, K W J (1975) PhD Thesis, Brunel University. 8 Hamid, M S, aind Ghista, D N (I974) Proceedings of the I974 International Conference on Finite Element Methods in Engineering held at the University of New South WVales, Australia (ed. Pulmano, V A, and Kabaila, A P), P. 337. Australia, Clarendon Press. 9 Evans, N T S, and Gourlay, A R journal of the Institute of Mathematics and its Applications. In press.
io Crank, J, and Gupta, R S (1972) Journal of the Institute of Mathematics and its Applications, I0, 19, 296. ii Pack, A I, Murray-Smith, D, Mills, R J, Hooper, M, and Taylor, J (I974) Bulletin of the Institute of Mathematics and its Applications, IO, 20 12 Zamir, M, and Roach, M R (I973) Journal of Theoretical Biology, 42, 33. 13 Crank, J (I974) Bulletin of the Institute of Mathematics and its Applications, 1o, 42.
