A DIRECTOR THEORY APPROACH FOR MODELING BLOOD FLOW IN THE ARTERIAL SYSTEM: AN ALTERNATIVE TO CLASSICAL 1D MODELS

Abstract

It remains computationally infeasible to model the full three-dimensional (3D) equations for blood flow in large sections of the circulatory system. As a result, one-dimensional (1D) and lumped parameter models play an important role in studies of the arterial system. A variety of 1D models are used, distinguished by the closure approximations employed. In this paper, we introduce a nine-director theory for flow in axisymmetric bodies as an alternative to the 1D models. Advantages of the director theory include (i) the theory makes use of all components of linear momentum; (ii) the flow is not assumed to be uni-directional; (iii) the theory is hierarchical; (iv) there is no need for closure approximations; and (v) wall shear stress enters directly as a dependent variable. In order to simplify the equations for mathematical analysis, for this work, attention is focused on cases where it is appropriate to model the flow as quasi-steady and the wall motion does not have a significant impact on bulk flow parameters. This work lays the foundation for future applications of the theory to unsteady flows in flexible walled vessels. For the geometries considered here, the nine-director theory has the same advantage as 1D models in providing a relatively simple relation between flow rate and average pressure drop. Conditions for existence, uniqueness and local stability of steady solutions are determined for both the 1D and nine-director equations. The predictive capability of classical 1D models found in the recent literature and a nine-director model7,15 are carefully evaluated through comparison with analytical and computational solutions to the axisymmetric, steady Navier–Stokes equations in geometries relevant to blood flow. For these benchmark problems over the range of Reynolds numbers considered, the nine-director theory is found to provide better results than the classical 1D models. A novel approach for parameter identification is in the 1D model is given and shown to substantially improve its predictive capability in these test cases.