Introduction

A famous orator or scientist from the nineteenth century once claimed that only the worst sort of mathematician would apply mathematics to medicine. With so many complicated relationships between all the variables, surely any mathematical model must become incoherent and incomprehensible? Surely this must be the case?

I think not. However, it has been my experience that mathematicians and medical people do not all talk on the same wavelength and this has probably much to do with their differing agendas. Medical researchers and practitioners usually have a concrete problem they want an answer to. They may then have recourse to mathematics to help find them solutions. Mathematicians, on the other hand, see medicine as a fountain of new and interesting mathematical problems leading them, perhaps, to research funding.

The Centre for Mathematical and Computational Science in Medicine (CMCSM) was set up as a joint initiative between the Universities of Glasgow and Strathclyde with a three year mandate from the Scottish Higher Education Funding Council to provide mathematical and computing science support to healthcare in Scotland. Our research here, whilst having a firm mathematical and computing skills base, has been of a far more practical nature than the research currently undertaken by the majority of Scottish mathematics departments engaged in modelling in medicine. At the CMCSM there are a number of research assistants each working in collaboration with medical researchers and companies from across Central Scotland. We have at present undertaken several projects, a sample of which includes identification of failing artificial heart valves from their phonocardiograms and in the computational issues associated with the secure remote transmission of medical data. I have been involved with two projects: a neuroimaging project which aimed to evaluate the amount of functional assymmetry in epilepsy patients, and in a renal graft simulation which I have been invited to discuss here.

My involvement in renal research began after a visit to our centre by my (now) principal collaborator, Dr.Alan Jardine of the the Western Infirmary Renal Unit and of the University of Glasgow's Department of Medicine and Therapeutics. Initially he had had some support from our Department of Statistics and had wondered if a new approach, provided by the CMCSM, could yield new insight into his renal transplant database beyond that typically afforded by the traditional suite of statistical tests. As a first step, some data-mining techniques were applied to the renal unit's database but no interesting results were obtained. I had initially suggested a compartmental differential equation model based on serum-creatinine clearance. However, this model required input from and produced results to variables we did not have easy experimental access. The dynamic time-evolution nature of the clearance model did, however, intrigue Alan and we began discussing whether or not his survivor functions could be used to drive a simulation of renal unit patient numbers. Since his survivor functions included cardiovascular risk profiles of patients, we would then be in a position to simulate transplant strategies aimed at special patient subgroups (such as the non-smokers, for example). Part I of this work will illustrate these simulations.

In Part II I discuss the progressive nature of renal disease. Unfortunately, after an extensive search of the literature I could find very little in the way of mathematical models of renal disease. However, one such model is discussed by Charturvedi & Insana (1997) [henceforth referred to as CI97]. They have developed a coupled system of differential equations linking the sclerosis index to glomerular hypertrophy. This model is unsatisfactory in many ways and I have developed my own system to which CI97 may be compared. I conclude this article with a brief summary and list of references.