Mathematical models and anaesthesia

The use of mathematics in medicine is not as widespread as it might be. While professional engineers are instructed in a wide variety of mathematical techniques during their training in preparation for their daily practice, tradition and the demands of other subjects mean that doctors give little attention to numerical matters in their education. A smattering of statistical concepts is typically the main mathematical field that we apply to medicine. The concept of the mathematical model is important and indeed familiar; personal finance, route planning, home decorating, and domestic projects all require the application of the basic mathematical tools we acquire at school. This utility is why we learn them. The insight that can be gained by applying mathematics to physiological and other problems within medical practice is, however, underexploited. The undoubted complexity of human biology and pathology perhaps leads us to give up too soon. There are useful and practical lessons that can be learned from the use of elementary mathematics in medicine. Anaesthetic training in particular lends itself to such learning with its emphasis on physics and clinical measurement. Much can be achieved with simple linear functions and hyperbolas. Further exploration into exponential and sinusoidal functions, although a little more challenging, is well within our scope and enables us to cope with many time-dependent and oscillatory phenomena that are important in clinical anaesthetic practice. Some fundamental physiological relationships are explained in this chapter using elementary mathematical functions. Building further on the foundation of simple models to cope with more complexity enables us to see the process, examine the predictions, and, most importantly, assess the plausibility of these models in practice. Understanding the structure of the model enables intelligent interpretation of its output. Some may be inspired to investigate some of the mathematical concepts and their applications further. The rewards can be intellectually, aesthetically, and practically fruitful. The subtle, revelatory, and quite beautiful connection between exponential and trigonometric functions through the concept of complex numbers is one example. That this connection has widespread practical importance too is most pleasing.