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. Its main function is to distribute oxygen-rich arterial blood to the cerebral mass. A 1-dimensional model of the CoW has been created to simulate a series of possible clinical scenarios such as occlusions in afferent arteries, absent or string-like circulus vessels, or arterial infarctions. The model captures cerebral haemodynamic auto-regulation by using a proportional-integral-derivative (PID) controller to modify efferent resistances and maintain optimal efferent flowrates for a given circle geometry and afferent blood pressure. Results match limited clinical data and results obtained in prior studies to within 6%. In addition, a set of boundary conditions and geometry is presented for which the auto-regulated system cannot provide the necessary efferent flowrates and perfusion, representing a condition with increased risk of stroke and highlighting the importance of modelling the haemodynamics of the CoW. The system model created is computationally simple so it can be used to identify at-risk cerebral arterial geometries and conditions prior to surgery or other clinical procedures.

A Model of Cerebrovascular Reactivity Including the Circle of Willis and Cortical Anastomoses

“…37,38,53,80 For instance, some models permitted clarification of several important issues of the intracranial circulation, such as the intracranial pressure (ICP) response to pressure-volume tests, 34 the significance of ICP and transcranial Doppler pulsatility indices, 71,72 the origin of ICP waves, 60,69,74 the characteristics of cerebral blood flow (CBF) regulation, 75 as well as the role of venous collapsibility. 12,68 Many models were focused specifically on fluidodynamics of the cerebral circulation, using 1D, 2D, or 3D partial derivative equations for the arterial blood flow, but almost completely neglecting the action of regulatory mechanisms, 2,7,8,10,11,18,42,54,80 or including blood flow regulation only in a simple form (for instance with a proportional-integral-derivative controller, such as done by Moorhead et al 50 ). These models are very useful to analyze mechanical aspects of the brain circulation (such as the mechanisms leading to aneurism rupture, the velocity profile in different arteries or pulse wave propagation in cerebral vessels) but are intrinsically unable to describe cerebral hemodynamics during transient maneuvers (such as hypercapnia, hypotension, and compression tests) which acutely activate control actions.…”

3D models of blood flow in the cerebral vasculature

“…These effects are of particular importance when investigating how blood is redistributed via the circle of Willis. It should be noted that there have been numerous cerebral autoregulation models proposed (Ursino and Giulioni 2003;Ursino 1991;Olufsen et al 2002), including models incorporated with a circle of Willis (Moorhead et al 2004;Ferrandez et al 2002;David et al 2003). However, the combination of autoregulation with the more realistic 3D circle of Willis geometries remains a relatively unexplored field.…”

“…There has been a significant body of research performed on blood flow in the circle of Willis (Moorhead et al 2004;Hillen et al 1986;Cassot et al 2000), treating the cerebral vasculature as a 1D structure and assuming Poiseuille flow. This approach however cannot capture the effects of the complex arterial geometry, in particular the effects of blood vessel junctions.…”

Multi-scale modeling on human intravascular cooling to induce brain hypothermia via circle of Willis

“…In order to understand how blood is perfused throughout the human brain, many scientists have investigated the effects of blood flow in the circle of Willis [16][17][18]. However, only a few attempted to predict the temperature field in the human brain using the blood flow in the circle of Willis.…”