This article mainly concerns with mathematical modelling in medicine. There is one area

in medicine namely pharmaco- kinetics which have already been mathematicized. There

are large areas in medical science which are not amenable to mathematical treatment,

and modelling constantly endeavour to widen the areas to which mathematical techniques

can be applied for gaining a better insight, and help deepen our understanding of those

areas which have already been mathematicized. The skills needed to be successful in

applying mathematics are quite different from those needed to understand concepts, to

prove theorems or to solve equations. The difficulty is not in learning and understanding

the mathematics involved but in seeing where and how to apply it. In this paper an

attempt is made to demonstrate the essentials of mathematical modelling without going

deep in to the details on specialised topics

Venkatesan, P

NIRT Institutional Repository

Biomedicine;1993(2)  15 - 18MATHEMATICAL MODELLING IN MEDICAL SCIENCESP. VENKATESANDepartment of StatisticsTuberculosis Research CentreMadras 600 034.AbstractThis article mainly concerns withmat ematical modelling in medicine. There is one areain medicine namely pharmaco- kinetics which have already been mathematicized. Thereare large areas in medical science which are not amenable to mathematical treatment,and modelling constantly endeavour to widen the areas to which mathematical techniquescan be applied for gaining a better insight, and help deepen our understanding of thoseareas which have already been mathematicized. The skills needed to be successful inapplying mathematics are quite different from those needed to understand concepts, toprove theorems or to solve equations. The difficulty is not in learning and understandingthe mathematics involved but in seeing where and how to apply it. In this paper anattempt is made to demonstrate the essentials of mathematical modelling without goingdeep in to the details on specialised topics.IntroductionMathematics is a powerful, flex&Language, models are representations andmathematical models are representationsframed in mathematical terms. A goodmetaphor for mathematical model is a map.Models are representations for a purpose andsome knowledge is required to build and usethem. Medical theories entail a concern withamong other things, generality, precision andparsimony.The knowledge required formathematical model building includes goodintuition and ability to abstract, someknowledge of a range of models, some facilitywith the manipulation of the mathematicalrepresentations, an ability to assess criticallythe models, some flair and modesty. In thisarticle, an attempt is made to give a briefintroduction to mathematical modelling to getan insight into the problems of medicalsciences.Early Expositions of theMethodlogy of ModellingMost of the early developments inmethodologies are due to physical sciences.Klamin (1971) proposed a five stage model ofthe problem solving process. The stages beingrecognition, formulation, solution, computationand explanation. Klamin described problemsolving as a linear process starting at stage oneand proceeding to stage five at which point theproblem is solved. Lin (1976) added theev luation of modelling process as sixth stage.In a ubsequent paper Lin (1978) addressed toanother important concept of a range ofmodels instead of a single best model.Wood (1969) described modellingprocess as an iterative process with evaluationof the results in the light of obsenable realityleading to modification of the model andrepetition of the stages - a form of parameteridentification problems. Hall (1972) further        developed the iterative methodology and Bajpaiet al (1975) used a flow block diagram toillustrate their concepts of methodologics ofmathematical modelling. A fu r therimprovement in this direction is due tod’Irveno and McLone (1977) where approachis to start by constructing the simplest model.If this is inappropriate then go back and makethe model more sophisticated usually bydropping or altering one or more of theassumptions. A novel treatment of modellingprocess by Penrose (1978) consists of six stagesin a circular progression His representation isreproduced in Fig. 1.The boxes are numbered and joined byarrows to indicate the normal direction oftravel from one to another, but it is usual to16 Biomedicine;1993(2) 15 - 18return to some of the boxes several-times asone’s ideas develop. A good account ofcomplex linkage methodology of mathematicalmodelling was given by Clements (1989).In this article model building is considered within the broad methodologieshaving epistemological, substantial, technicaland purposivecomponents. Within theepistemological component, criteria for theconstruction and evaluation of the models areset forth. Within the substantial components thenature of theories, its theoretical terms and thenature of relevant data are formulated. Thetechnical component deals with the nuts andbolts of modelling - the mathematical systemsused, their properties, measurement, andestimation. Within the purposive component,we discuss the goals of the modelling effort.These include explanation, description andexplication. They also include a considerationof the uses of a model. These components arehighly interrelated.Classification of models : Mathematical modelsare frequentlyco-ordinated by threedistinctions: The first differentiates, processmodels from structural models, the seconddeterministic models from probabilistic models,and the third, using discrete from continuous.In principle, an eight-cell table can be constructed and each cell fitted withmathematical models sharing the criteriadefining the cell.understand the structure of systems relations.The tools used include graph theory, matrixalgebra, groups, semi groups., Boolean algebra,algebraic topology. Statistical models are usedto model processeswhose outcomes aregoverned by a probabilistic mechanism(s).Deterministic models eschew stochasticmechanisms infavour of deterministicmechanisms and relations. Process models maybe deterministic or stochastic, but structuralmodels tend to be deterministic.The distinction between mathematicalmodels and statistical analysis is a very blunedone. Formal approaches can and do   incorporate error specification which otherapproaches generally do not. This informsestimation. Second, the properties of thestatistical tools are stated and establishedmathematically. Finally, new mathematicalmodels and their uses generate estimationproblem and statistical questions.The Theory, Model and Data Mangle : Thistriangle provides a method for selecting modelsbest suited for a substantive problem whereskill shows alternative model candidate6 andselection of most. fruitful models. There arethree pairs of mappings - between theory andmodel, between model and data and betweentheory and data and all are important.The theory -model linkage isconcerned with expressing a congruencebetween a theory and its representation in amathematical model The theory has to mapinto the model with little distortion or loss.Deductively, there is a mathematicalformalization of the theory while inductivelythis can be a formal generalizaton of thetheory. The mathematical model then has to beuseful. These results can be mapped to theoryand data. Deductively model maps to data byspecifying or predicting empirical outcomes.Also deductively mathematical results map totheory by specifying theoretical implications ofthe derivations through mappings linking theoryand model. The theory, the model and the datahave to make sense and be consistent with oneProcess models explicitly, attempt tomodel the changes andprovide ananother - which is the nub of evaluating modelsunderstanding of the mechanisms of change.(Doreian, 1987).Among the frequently used tools aredifferential equations and difference equations.Metaphor, Analogy and Model: There are noStructural models attempt to represent andclear cut simple definitions for the terms like   metaphor, analogy and model, but althoughBiomedicine;1993(2) 15 - 18 17they are by no means synonyms, they do sharecertain features. Broadly speaking it can besaid that they all attempt to enhance ourunderstanding of new information in terms of what is already familiar. According to Sutton(1978), "Analogies are extended similes inwhich an attempt is made to trace multiplepoints of comparisons, Metaphors arc ... lessexplicit and much more mentally teasing ...forcing the hearer to search among hisassociated ideas for possible connections and,a model can be thought of as an extendedmetaphor". conc ntration isproportional to theconc ntration of the drug in the blood stream.In mathematical terms(3.3)where k is a positive constant. Experimentshave. shown that (33) is a good approximationto reality for many drugs, the most importantbeing penicillin and streptomycinHaving determined the constant k fora particular drug we now use equation(3.3) asAn Applications of Modelling withDifferential Equationsa model. Suppose the patient is given an initialdose yo, which is assumed to be instantaneouslyabsorbed by blood at time t = 0 resulting inrepresents a vast variety of situations in medicalsciences. Problems in population growth, drugabsorption and elimination, alcohol absorption,water cooling carbon dating etc can bemodelled by equations of type (3.1). Once thisdifferential equation has been solved we haveeffectively solved numerous problems. Thesolutions to the above problem can beexpressed in the formy = yo exp (kt) (3.2)-k showing that the drugs’ concentration decaysexponentially. After a fixed time T, a seconddose yo is administered. Just before the dose,the amount in the blood is given byJust after the second dose, at timeT = T+where yo is a initial value. The behaviour of the = yo (1+exp(-kT)) (3.6)solution depends on the sign of the constant k.If k is positive we have exponential growth, if The new quantity decays according tok is zero, y remains equal to its initial value,law (33) with initial conditionand if k is negative we have exponential decay,y = yo (1 + exp(-kT)) at t = TThe study of the way in which a drug Thus for t > Tloses its concentration in the blood of a patientis fundamental to pharmacology, The'dose-response’ relationship plays a vital role inAgain giving the patient a dose yo at t = 2Tdetermining the required dosage level and the      results in interval of time between doses for a particulardrug.y ( 2T + )Suppose y= y(t) represents the (3.8)quantity of drug in the blood stream at time t.The simplest way to model such behaviour isassume that the rate of change of theAgain solving (33) with18 Biomedicine;1993(2) 15 - 18y = yo (1+exp(-kT)+exp(-2kT))at t > 2T gives(3.9) y(t) = yo (1 + exp(-kT) + exp (-2kT)exp(-k(t-2T))      (3.10) Continuing this wayy(nT+) =yo (1 +exp(-kT) + .... +exp(-nkT) (3.11)for n=1, 2, . . . . .y(nT+) = yo (1-exp(-(n+1)kT))/(1-exp(-kT))  (3.12)As n gets larger, exp(-(n+1)kT -> 0 so thaty(nT+) = yo / (1-exp(-kT))Since this is independent of n, the modelpredicts that the quantity of drug is tending toa saturation level say Ys, whereYs = yo / (1-exp(-kT)) This formula can be used fordetermining (i) the required time interval T,between doses for a given dose yo andprescribed final level Ys. (ii) the dose level yorequired to obtain a final dose level Ys with aprescribed interval between doses, T.DiscussionSince the situations in medicalsciences are quite complex, one should havesome insight into the problem before heattempts to formulate a new mathematicalmodel. A good review in this aspect has beengiven by Kapur (1985). Once a model isformulated  its consequence can be deduced byusing mathematical techniques and the resultscan be compared with observations. Thediscrepancies between theoretical considerationand observations suggest further improvementin the model as suggested by Clements (1989)and Penrose (1978). This process is repeatedtill a readily satisfactory model is obtained. Itis important to realise that learning to applymathematics is a very different activity fromlearning mathematics. There is no theory tolearn mathematical modelling and there areonly a few guiding principles. There are manyexamples of very simple mathematics givinguseful solutions to very difficult problems inbiomedicine, although generally speaking thecomplexity of the problems and the requiredmathematical treatment go hand in hand Thereis growing fraternity of biomedical scientistsinvolved in mathematical modelling now. Manyjournals cater to a large extent for the needs ofmathematical modelling and three of themneed mention are : (i) Applied Mathematicalmodelling (ii) International Journal ofMathematical model&g and (ii) InternationalJournal of Mathematics and Computersimulations. To conclude, we quote“All knowledge is, in the final analysis,History,All Sciences are, in the abstract,Mathematics’.Referencesl.2.3.4.5.6.7.8.9.10.11.12.Bajpai, AC, Mustoe LR and Walter, D. (1975).Mathematical education of engineers. Part L Acritical appraisal, Int. J. Math. Educ Sci. Technol., 6:361-65.Clements, D. (1989). Mathematical modelling : Acase study 1 approach Cambridge University Press,Cambridge.d’Invcrno, RA aid Mclonme, RR. (1977). Amodelling approach totraditional appliedMathematia. Math. Gaz. 61: 92-95.Doreian, P. (1987). Mathematical Modelling : InMethods, Ethics and Modek. (Ed. J. Kuper). P.70-76Routledgc and Keganpaul, N.Y.Hall, G.G. (1972). Modelling a phi for appliedmathematics. Bull IMA, 8: 226-31.Klamin, M. (1971): On the ideal role of anmathematician and its educational implitcations. Am.Math. Monthly, 78: 53-58.Lin, CC (1976). On the role of applied mathematics.Advances in Math, 19: 267-72.Lin, CC (1978). Education of applied mathematicianSIAM Review, to: 838-45.Kapur, JH. (1985). Mathematical modelling inbiology and medicine. Affiliated East-West Press. N.Delhi.Penrose, O. (1978). How can we teach mathematicalmodelling ? J. Math. Model. Teachers, 1: 31-36.Sutton, C (1978). Metaphorically speaking. L&esterUniversify School of Education, London.Woods, IX (1969). What is wrong with appliedMathematics, Bull., IMA, 5: 70-74.

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