Damping mechanisms and shock-like transitions in the human arterial tree

Abstract

The unidirectional nature of blood flow in mammalian arteries encourages the modelling of propagating pressure and flow pulses in the arterial tree by means of a one-dimensional mathematical approximation. It has been shown that this approximation yields realistic results in the proximal as well as in the distal regions of a simulated arterial conduit, providing that the damping induced by the viscoelasticity of the vessel walls as well as by the viscosity of the blood are properly taken into account. Often, models were formulated on the basis of an elastic formulation of the arterial wall properties and of an approximation for the damping due to blood viscosity which is derived from parabolic velocity profiles (“Poiseuille model”). Yet, such models are known to produce shock-like transitions in the propagating pulses which are not observed in man under physiological conditions. The viscoelastic damping characteristics of the vessel walls are such that they reduce the tendency of shock formation in the model. This can be shown with the aid of a wave front expansion, from which criteria for the steepening of wave fronts are derived. The application of the results to the human arterial system shows that shock-like waves are not to be expected under normal conditions. However, in case of a pathologically increased pressure rise at the root of the aorta, shock-like transitions may still develop in the periphery. Such circumstances can occur, e. g., in cases of severe aortic valve insufficiency (without aortic stenosis). Furthermore, the damping characteristics associated with blood viscosity are shown to impede mathematical shock formation in the strict sense a priori. Steepening of a wave front according to the criteria mentioned above still may occur, however, with increasing steepening this process is gradually counterbalanced by the dissipative mechanism induced by viscous friction. Finally, the relative influence of the damping due to wall viscoelasticity and blood viscosity is discussed as a function of the relevant parameters describing the geometry and pulsatility of the typical blood flow conditions.