Rheological approaches of arteries

A fundamental problem of haemodynamics lies in the description of the rheological properties of arteries. The time and history dependency of stress and strain, the nonlinearity of the stress-radius relationship, and the activity of vascular smooth muscle complicate or even prevent a complete mathematical characterization of the arterial wall mechanics. Due to this nonlinearity, dynamic investigations were hitherto performed in excised arteries in vitro by means of small sinusoidal changes of stress and radius at different stress levels in a wide frequency range. To allow an analysis of the dynamic rheological properties of arteries in vivo, we have developed a procedure which permits the separate determination of the elastic, the viscous, and the inertial forces acting on the arterial wall. The stress can be subdivided into an elastic stress which is a function of radius (r), a viscous stress which is a function of dr/dt, and an inertial stress which is a function of d2r/dt2. These stresses are formulated as polynomials. Under cyclic loading and unloading, hysteresis loops appear in the stress-radius diagrams of arteries. Since the elastic stress-radius diagram must be free from any loop, the coefficients of the viscous and the inertial stress can be found by a fitting procedure, using the criterion of loop elimination. Investigations were performed on exposed canine arteries in vivo. The main result was that the elastic stress-radius curve was markedly nonlinear at greater pulse pressures. The viscous wall behaviour, too, was nonlinear and depended mainly on the square of the vessel radius.