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

The authors have designed a mathematical model to investigate the influence s of the physical and chemical properties of the cerebral blood vessel resi stance 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 a ffecting the vascular smooth muscle cells and the vascular walls. In partic ular, we shall model a series of calcium sites and derive a generalized equ ation of the diameter as a function of pressure. The model includes the act ion 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 alt erations to their properties within the blood vessel. They are formulated i n terms of three parameters: the diameter at zero pressure, the myogenic re sponse as the pressure tends to zero and a term associated with the myogeni c tone. All three parameters may be reliably extracted from diameter-pressu re measurements. The model was successfully used in quantifying diameter os cillations and dynamic myogenic responses that are frequently observed both in vivo and in vitro. Finally, we tested the model on experimental data ob tained from the resistance of cerebral vessels that have been isolated from rats. In particular, we have first shown that the blood vessel characteris tics 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 diamete r.