Mathematical modelling of arterial fluid dynamics

Two major research themes have dominated the fluid dynamical study of blood flow in arteries: ( a) propagation of the pressure pulse and (b) flow patterns and wall shear stress (WSS) distribution in complex arterial geometries. The former led to physiological understanding and permitted the interpretation of diagnostic measurements of the wave-forms of blood pressure and flow-rate, for example. The latter was driven by the need to understand the link between wall shear stress and the development of arterial disease, and the understanding gained is also used in the design of surgical interventions such as bypass grafts. Pulse wave modelling has always been essentially mathematical, using one-dimensional linear or weakly nonlinear theory, and can therefore give significant understanding very simply, as is outlined in this paper. The relatively new wave-intensity analysis of the pulse wave shows that the subject is still capable of giving new insight. The study of time-dependent flow in complex three-dimensional geometry, even when the tubes are taken to be rigid and the fluid Newtonian, is much more difficult. Realistic simulation requires the computational solution of the full Navier-Stokes equations, in a geometry obtained from a particular subject by means of magnetic resonance imaging ( say), using input flow or pressure data that are also obtained by non-invasive imaging. The combined computational procedure has not yet been developed to the point at which one can have confidence in its accuracy, but it soon will be. However, this is not mathematical modelling and does not clearly lead to new fluid dynamical understanding. For that one must go to idealised models such as uniform curved tubes, which lead to interesting fluid dynamics, but it is not clear how relevant they are to biomedical practice. To show that mathematical modelling is not dead, the paper will conclude with a brief description of a recent model of the new process of transmyocardial laser revascularisation, developed to restore oxygen supply to heart muscle cure off by an infarct, for example.