The propagation of pressure and velocity waves in liquid-filled elastic tubes has often been the subject of careful analysis [1,2] as a model of the behavior of pulse waves in arteries. Practical requirements often make it necessary to return to these theories in order to clarify various details. Below, we examine one such problem generated by the proposal to use the amplitude characteristics of the pulse wave as a medical diagnostic index [3,4] rather than the pulse wave propagation characteristics, as is usually the case.
FORMULATION OF THE BASIC PROBLEMIn experiments it was long ago observed that when an organ (for example, the muscles of the lower extremity in man) are working hard, the resistance of the microvascular bed falls as a result of regulatory mechanisms cutting in, and that subsequently the flow in the main arteries supplying the organ with blood is restrucatred: the blood flow increases and the shape of the pulse wave changes [5--7]. Usually, there is an increase in the diastolic (i.e., minimum over the cardiac cycle) value of the velocity u a. For the muscles of the extremity it is negative when they are at rest and positive when they are working ( Fig. 1, curves 1 and 2, Uo=U(X=0)).It was recently proposed [3,4] that the diastolic velocity u a be used as a diagnostic index, since it is sensitive to changes in the functioning of the heart and the state of the vessels above and below the point at which the velocity is measured. There is reason to believe that the dependence of ua on how hard the muscles of the extremity are working contains additional useful information.In connection with this proposal, it has become necessary, firstly, to understand the mechanism of variation of the diastolic velocity and, secondly, to interpret its dependence on the blood flow conditions, the state of the vessels, etc.The qualitative answer to the first question is very simple: with variation in the output resistance the mean blood flow through the artery over the pulse cycle varies within wide limits in inverse proportion to that resistance, while the amplitude of the flow rate oscillations, determined by the impedance of the artery and the condition of wave reflection from the output resistance, may vary by not more than a factor of 2 [2]. Consequently, theoretically it is always possible to reduce the resistance to a point at which the difference between the mean value and the amplitude of the flow rate becomes positive.In order to answer the second question it is necessary to resort to a mathematical model of the motion of the blood along the artery.Let us consider an arterial vessel of circular cross section with a thin elastic wall (Fig. 2) having in equilibrium a constant cross-sectional area F o at the pressure Po-The vessel is connected with input and output resistances assumed, for simplicity, to be purely resistive; the length of the vessel L and the values of the resistances Z+ and Z_ are assumed to be constant. The pressure p+(t) > 0 at the input to the system is a given function with period T; the ...
