A One-dimensional Model of Blood Flow in Arteries with Friction and Convection Based on the Womersley Velocity Profile

Abstract

In this paper, we present a one-dimensional model for blood flow in arteries, without assuming an a priori shape for the velocity profile across an artery (Azer, Ph.D. thesis, Courant Institute, New York University, 2006). We combine the one-dimensional equations for conservation of mass and momentum with the Womersley model for the velocity profile in an iterative way. The pressure gradient of the one-dimensional model drives the Womersley equations, and the velocity profiles calculated then feed back into both the friction and nonlinear parts of the one-dimensional model. Besides enabling us to evaluate the friction correctly and also to use the velocity profile to correct the nonlinear terms, having the velocity profile available as output should be useful in a variety of applications. We present flow simulations using both structured trees and pure resistance models for the small arteries, and compare the resulting flow and pressure waves under various friction models. Moreover, we show how to couple the one-dimensional equations with the Taylor diffusion limit (Azer, Int J Heat Mass Transfer 2005;48:2735-40; Taylor, Proc R Soc Lond Ser A 1953;219:186-203) of the convection-diffusion equations to drive the concentration of a solute along an artery in time.