Mathematical modelling of responses of cerebral blood vessels to changing intraluminal pressure

The authors have designed a mathematical model to investigate the influences of the physical and chemical properties of the cerebral blood vessel resistance on vessel diameter. The model is based on the way the total tension within the blood vessel walls varies due to specific ions interacting and affecting the vascular smooth muscle cells and the vascular walls. In particular, we shall model a series of calcium sites and derive a generalized equation of the diameter as a function of pressure. The model includes the action of the vascular smooth muscle cells and the elasticity of the vascular walls, the pressure exerted on the walls by the blood and the effect of alterations to their properties within the blood vessel. They are formulated in terms of three parameters: the diameter at zero pressure, the myogenic response as the pressure tends to zero and a term associated with the myogenic tone. All three parameters may be reliably extracted from diameter-pressure measurements. The model was successfully used in quantifying diameter oscillations and dynamic myogenic responses that are frequently observed both in vivo and in vitro. Finally, we tested the model on experimental data obtained from the resistance of cerebral vessels that have been isolated from rats. In particular, we have first shown that the blood vessel characteristics are such that the diameter change due to calcium ion variations is at a maximum value. Second, we have shown that blood flow affects the myogenic response and third, we can explain the affect of ATP on the vessel diameter.