Examining the response pressure along a fluid-filled elastic tube to comprehend Frank's arterial resonance model

Frank first proposed the arterial resonance in 1899. Arteries are blood-filled elastic vessels, but resonance phenomena for a fluid-filled elastic tube has not drawn much attention yet. In this study, we measured the pressure along long elastic tubes in response to either a single impulsive water ejection or a periodic water input. The experimental results showed the low damped pressure oscillation initiated by a single impulsive water input; and the natural frequencies of the tube, identified by the peaks of the response in the frequency domain, were inversely proportional to the length of the tube. We found that the response to the periodic input reached a steady distributed oscillation with the same period of the input after a short transient time; and the optimal pressure response, or resonance, occurred when the pumping frequency was near the fundamental natural frequency of the system. We pointed out that the distributed forced oscillation could also be a suitable approach to analyze the arterial pressure wave. Unlike Frank's resonance model in which the whole arterial system was lumped together to a simple 0-D oscillator and got only one natural frequency, a tube has more than one natural frequency because the pressure P(z,t) is a 1-D oscillatory function of the axial position z and the time t. The benefit of having more than one natural frequency was then discussed.