A Mathematical Model for Autoregulation of the Arterial Lumen by Endothelium-Derived Relaxing Factor

Local regulation of blood flow and blood pressure in the cardiovascular system is provided by a complex network of various mechanical and biochemical factors. Fluid shear stress is one of the important regulators mediated via the endothelium-derived relaxing factor (EDRF) that is nitric oxide. This mechanism involves biochemical reactions in the arterial wall. The autoregulation process is managed by the vascular tone and gives a negative feedback to the shear stress change. A new mathematical model for the autoregulation of the arterial lumen under constant transmural pressure is presented. The model takes account of EDRF, the multi-layer structure of an arterial wall, its nonlinear viscoelastic properties, and diffusion-kinetic processes. The exact stationary distributions of NO, Ca²⁺ and phosphorylated myosin in an arterial wall are found. The condition of linear stability for equilibrium state is obtained analytically. Numerical simulation for the autoregulation process is presented. It is shown that there is limit cycle bifurcation at some wall viscosity values outside linear stability range. The transition of the system to a new equilibrium state with a delay in response to the change in a blood flow rate is demonstrated. Slow oscillations of about 0.01 Hz managed by muscle tone are observed. The results are in good agreement with the experiments of Snow, et al. and Melkumyants, et al. carried out in canine and feline arteries.