Lumped Parameter and Feedback Control Models of the Auto-Regulatory Response in the Circle of Willis

Abstract

The Circle of Willis (CoW) is a ring-like structure of blood vessels found beneath the hypothalamus at the base of the brain, which distributes blood to the cerebral mass. Simple models of cerebral blood flow dynamics would create a tool capable of diagnosing potential outcomes of surgical or other therapies.

A one-dimensional flow model is developed to capture the auto-regulation dynamics by which cerebral blood perfusion is maintained. Figure 1 shows a CoW schematic composed of circulus, afferent (inflow) and efferent (outflow) arteries. Positive flow around the CoW is clockwise but not restricted in direction, while flow in afferent and efferent vessels is restricted to the directions shown. Flow in efferent vessels is regulated by time varying peripheral resistances modelling the effects of auto-regulation in the smaller blood vessels they supply. While many individuals have complete and symmetrical CoW geometry, it is not uncommon for some elements to be restricted or omitted (van der Zwan and Hillen 1991) with the communicating vessels having a higher occurrence of omission (Alpers and Berry 1963).