A single-tube mathematical model of reactive hyperaemia

A mathematical model of reactive hyperaemia is developed using quasi-steady flow in a single tube to represent blood flow in the vascular bed. The role of the myogenic response during reactive hyperaemia is examined by suggesting a linear relationship between tube cross-sectional area S and pressure p, in which S decreases as p increases, thereby modelling the response of the smooth muscle in the blood vessel walls to increases in p which the myogenic mechanism proposes. However, this simple relationship, together with the equations of continuity and Poiseuille flow, lead to an unstable equation for p which is inconsistent with the known boundary conditions. It is necessary to make S a function of p and delta p/delta t in order to achieve a stable response which implies that the myogenic response must be rate sensitive to pressure changes. The resulting equations are then solved for p, S, and flow Q by numerical integration and give results for Q which are in broad agreement with experiment. The model also suggests that the changing pressure gradient governs the flow in reactive hyperaemia rather than changes in the resistance of the blood vessels.