A branched one-dimensional model of vessel networks

We introduce a model representing the venous network of the leg. The network consists of a coupled system of elastic tubes. The flow through each elastic tube is assumed to be unsteady, incompressible and one-dimensional. The network topology, as well as the lengths and diameters of the tubes, is based on literature data. As in the human leg the network is composed of two sub-networks, deep and superficial, which are connected by transverse segments. We introduce a new model for confluences or branching points, as well as models of the valvular system and of the muscular activity. We perform a numerical study of the transmission and reflection of waves at a confluence. Our model valvular system imposes a privileged direction of the flow towards the heart. Muscular activity is modelled using a modification of the tube law of the vessel and through an inflow of blood when muscle contraction pushes blood from the microcirculation to the veins. The model is capable of simulating several motions such as walking, dorsal flexion and tiptoe. Numerical tests show the physical relevance of the model, and in particular demonstrate that when the system is excited at the foot level, a two-frequency response appears. These frequencies are closely related to the characteristic lengths of the typical segments of the deep and of the superficial networks. We find good qualitative agreement between experimental and numerical flow rates, using clinical data corresponding to a single ‘tiptoe’ motion. We make numerical predictions of the internal venous pressure at the foot level in a valvular-incontinent system which agree with clinical observations.