Mathematical modelling in nuclear medicine

Modern imaging techniques can provide sequences of images giving signals proportional to the concentrations of tracers (by emission tomography), of X-ray-absorbing contrast materials (fast CT or perhaps NMR contrast), or of native chemical substances (NMR) in tissue regions at identifiable locations in 3D space. Methods for the analysis of the concentration-time curves with mathematical models describing the physiological processes and the appropriate anatomy are now available to give a quantitative portrayal of both structure and function: such is the approach to metabolic or functional imaging. One formulates a model first by defining what it should represent: this is the hypothesis. When translated into a self-consistent set of differential equations, the model becomes a mathematical model, a quantitative version of the hypothesis. This is what one would like to test against data. However, the next step is to reduce the mathematical model to a computable form; anatomically and physiologically realistic models account of the spatial gradients in concentrations within blood-tissue exchange units, while compartmental models simplify the equations by using the average concentrations. The former are known as distributed models and the latter as lumped compartmental or mixing chamber models. Since both are derived from the same ideas, the parameters are usually the same; their differences are in their ability to represent the hypothesis correctly, quantitatively, and sometimes in their computability. In this essay we review the philosophical and practical aspects of such modelling analysis for translating image sequences into physiological terms.