A One-Dimensional Fluid Dynamic Model of the Systemic Arteries

The systemic arteries are modeled as a bifurcating tree of compliant and tapering vessels. Blood flow and pressure in the vessels are determined by solving the axisymmetric Navier-Stokes equations. The arterial tree ranging from the aorta to the arterioles consists of a tree with more than 20 generations. Computing blood flow and pressure for all vessels requires a prohibitive amount of time. To avoid using too much time, we have truncated the arterial tree after a limited number of generations and applied a suitable outflow boundary condition. To this end we propose a structured tree model in which a root impedance is determined using a semi-analytical approach. The structured tree is a binary asymmetric tree in which the radii of the daughter vessels are scaled linearly with the radius of the parent vessel. The root impedance of the structured tree is found by propagating solutions of a wave equation from the terminals to the root of the structured tree. The wave equation is derived by linearizing the axisymmetric Navier-Stokes equations together with applying a long-wave approximation. The root impedance of the structured tree provides a dynamical outflow boundary condition, which is computationally feasible. The structured tree outflow boundary condition is based on physiologic principles and it accommodates wave propagation effects for the entire systemic arterial tree. Blood flow in the large systemic arteries is verified by comparing simulations with data obtained from magnetic resonance measurements. The outflow boundary condition is verified by comparisons with literature data and with a standard model (the three-element windkessel model).