Analysis of flow and vascular resistance in a model of the circle of Willis

A very simple model of the flow in the circle of Willis is described in this paper. Disregarding pulsatility and vessel wall elasticity, fluxes in all segments of the circle of Willis and its afferent and efferent vessels are calculated by applying the Poiseuille-Hagen formula. Comparison with the fluxes calculated numerically from a more sophisticated mathematical model, including pulsatility, vessel wall elasticity and nonlinear effects, revealed only very slight differences. In short, fluxes in the afferent vessels and the segments of the circle of Willis are influenced by any change of resistance within the network, whereas the fluxes in the efferent segments are dominated by the efferent resistance distribution. However, a great advantage of the present simple model is that it offers the possibility of an analytical approach which yields both an easy sensitivity analysis of parameters and an insight into the mechanisms that govern the flow in a network like the circle of Willis. It can be concluded that these mechanisms are similar to the principles of the Wheatstone bridge, known from electrical circuit theory.