Case studies have long been used to support the mathematics education of undergraduate engineers. Changes in the mathematical ability of entrants to engineering programmes and, indeed, the changing nature of many of the programmes themselves indicate the need to make the students' mathematical experiences more 'user friendly'.We describe here an approach which uses case studies, not as illustrations of applications of mathematics after a mathematical topic has been discussed, but in a fully integrated central role as vehicles for whole group discussion from which the students 'discover' the necessary mathematics which is taught subsequently.Not only is the 'carrot' of the application then central to their learning, but the need for the mathematics being taught is also clearly demonstrated. This approach has been tried with a group of 50 rst year engineers. The effects on student motivation, ability and knowledge retention are discussed together with an indication of the Integrated Case Studies which were used.

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SynopsisThe transport of most foreign substances through the membranes of living organisms is via a partitioning mechanism. Many studies have shown how biological potency varies with lipophilicity within a congeneric series, and several models have been proposed to explain this variation. Some of these models are based on equilibrium considerations, which seems hardly realistic for most applications in which a single dose is administered, absorbed and eventually excreted. Of the models available, that of Penniston et al.1 is favoured as being the most flexible and amenable to development. These authors used the model simply to show that, under certain prescribed conditions, the concentration of drug reaching the 'receptor', at a fixed time after dosage, varied approximately parabolically with lipophilicity ; this appeared to confirm many experimental observations that biological potency varied parabolically with lipophilicity. The model can, however, be used to indicate how response varies with time after dosage, and this paper considers the significance of such variation.The model consists of 19 alternate aqueous (odd-numbered) and lipid (even-numbered) compartments ( Figure 1). The coefficients k and I are the forward and reverse partitioning rate constants I 2 18 19 Compartment numberFigure 1. Model used to determine how response varies with time after dosage. k, Forward partitioning rate constant; I, reverse partitioning rate constant; m, constant.respectively from aqueous to lipid phase. The partition coefficient (P) is thus defined as k / l and for simplicity k x 1 is taken to be constant and equal to unity. This is not strictly true,2 but any variation of k x I with lipophilicity probably has only minor effects on the 'structure-activity' curves generated by the model. The coefficient m is a constant, irrespective of lipophilicity, and it will be seen that compartment 19 is a total sink, representing excretion and/or metabolism to an inactive species.Generally, unit dose is administered to compartment one at zero time.Differential equations are set up describing the rate of transfer between adjacent compartments, and these are solved by computer to give concentrations in each compartment as a function of a Presented at the symposium on Insecticide kinetics and time related effects in insecticide action organised by the

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