A Circle of Willis Simulation Using Distensible Vessels and Pulsatile Flow

[+] Author and Article Information
R. H. Kufahl, M. E. Clark

Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, Ill. 61801

J Biomech Eng 107(2), 112-122 (May 01, 1985) (11 pages) doi:10.1115/1.3138531 History: Received January 19, 1984; Revised July 10, 1984; Online June 15, 2009


The development of a one-dimensional numerical (finite-difference) model of the arterial network surrounding the circle of Willis is described based on the full Navier-Stokes and conservation of mass equations generalized for distensible vessels. The present model assumes an elastic wall defined by a logarithmic pressure-area relation obtained from the literature. The viscous term in the momentum equation is evaluated using the slope of a Karman-Pohlhausen velocity profile at the vessel boundary. The afferent vessels (two carotids and two vertebrals) are forced with a canine physiologic pressure signature corresponding to an aortic site. The network associated with each main efferent artery of the circle is represented by a single vessel containing an appropriate amount of resistance so that the mean flow through the system is distributed in accordance with the weight of brain irrigated by each vessel as determined from a steady flow model of the same network. This resistance is placed a quarter wave-length downstream from the heart to insure proper reflection from the terminations, where the quarter wavelength is determined using the frequency corresponding to the first minimum on an input impedance-frequency diagram obtained at the heart. Computer results are given as time histories of pressure and flow at any model nodal point starting from initial conditions of null flow and constant pressure throughout the model. Variations in these pressure and flow distributions caused by the introduction of pathologic situations into the model illustrate the efficacy of the simulation and of the circle in equalizing and redistributing flows in abnormal situations.

Copyright © 1985 by ASME
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