The natural frequencies of the arterial system and their relation to the heart rate

We assume the major function of the arterial system is transporting energy via its transverse vibration to facilitate the blood flowing all the way down to the microcirculation. A highly efficient system is related to maintaining a large pressure pulse along the artery for a given ventricular power. The arterial system is described as a composition of many infinitesimal Windkessels. The strong tethering in the longitudinal direction connects all the Windkessels together and makes them vibrate in coupled modes. It was assumed that at rest condition, the arterial system is in a steady distributed oscillatory state, which is the superposition of many harmonic modes of the transverse vibration in the arterial wall and the adherent blood. Every vibration mode has its own characteristic frequency, which depends on the geometry, the mass density, the elasticity, and the tethering of the arterial system. If the heart rate is near the fundamental natural frequency, the system is in a good resonance condition, we call this "frequency matching". In this condition, the pulsatile pressure wave is maximized. A pressure wave equation derived previously was used to predict this fundamental frequency. The theoretical result gave that heart rate is proportional to the average high-frequency phase velocity of the pressure wave and the inverse of the animal body length dimension. The area compliance related to the efficiency of the circulatory system is also mentioned.