Lumped Parameter and Feedback Control Models of the Auto-Regulatory
Response in the Circle of Willis
KT Moorhead
University of Canterbury, Christchurch, New Zealand
CV Doran
University of Canterbury, Christchurch, New Zealand
JG Chase
University of Canterbury, Christchurch, New Zealand
T David
University of Canterbury, Christchurch, New Zealand
Introduction
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 autoregulation 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).
Abbreviation
RRMCA
BA
RRPCoA
RRACA1
RRACA2
LPCA1
Left proximal Posterior Cerebral Artery
LPCoA
Left Posterior Communicating Artery
LICA
RRPCA1
RRICA
RBA
RACoA
+ve
RLICA
RLACA2
RRPCA2
RLACA1
RLPCoA
RLPCA2
RLMCA
Left Internal Carotid Artery
LACA1
Left proximal Anterior Cerebral Artery
ACoA
Anterior Communicating Artery
RACA1
Right proximal Anterior Cerebral Artery
RICA
RLPCA1
Vessel Full Name
Basilar Artery
Right Internal Carotid Artery
RPCoA
Right Posterior Communicating Artery
RPCA1
Right proximal Posterior Cerebral Artery
LPCA2
Left distal Posterior Cerebral Artery
LMCA
Left Middle Cerebral Artery
LACA2
Left distal Anterior Cerebral Artery
RACA2
Right distal Anterior Cerebral Artery
RMCA
Right Middle Cerebral Artery
RPCA2
Right distal Posterior Cerebral Artery
Figure 1: CoW Schematic Resistance Model showing flow directions and the
Abbreviations Used
1
Method
The CoW is modelled with laminar, viscous and incompressible flow assumptions,
with individual arteries being modelled by simple Poiseuille flow in a tube. The
relationship can also be represented using an electrical circuit analogy.
q
P1  P2
,
R
(1)
where P1-P2 is the pressure change across the artery, and R represents the arterial
resistance to blood flow, q. Using Poiseuelle flow assumptions, the resistance, R, can
be defined as a function of the artery length, l, artery radius, r, and dynamic viscosity
of blood, μ.
R
8l
,
r 4
(2)
The afferent and circulus arteries are modelled with constant resistances, and the
efferent arteries are modelled with variable resistors, as shown in Figure 1. Efferent
arteries are limited to changes in radius of up to 40% with the following limit on
resistance.
(1  0.95) R ref  R  (1  0.95) R ref
(3)
The auto-regulation process can be described by a typical Proportional-IntegralDerivative (PID) feedback control system, where the error between desired and actual
flowrates drives changes in resistance.
u(t )  K p e  K i  edt  K d
de
dt
(4)
where e is the error in the flowrate represented by e=(qref – q), and Kp, Ki and Kd are
control gains. The peripheral resistance dynamics are therefore described by a first
order system, with a driving input, u(t).
R  ( R  Rref )  u(t )
(5)
where  is the time constant of the auto-regulatory response, and R and Rref are the
actual and reference resistances of the efferent artery respectively. Equation (5) can
be used with Equations (1) and (4) to create a non-linear system.
A( x ) x (t )  b(t )
(6)
where A(x) is a matrix containing constant resistances for each arterial element
including those obtained from Equation (5), x(t) is a state vector containing timedependent flowrates through those arteries and nodal pressures at the end of the
arteries, and b(t) is a vector containing time-dependent arterial and venous pressures.
The basic data for length, and radius (Hillen et al. 1988), is used to calculate arterial
resistance using Equation (2), which are then normalised to the LICA. Resistance in
the ACA2, MCA, and PCA2 is set to approximate the ratio 6:3:4 in the asymptotic
flow state (Hillen et al. 1986).
The dynamic behaviour of the peripheral resistance enables recalculation of the
resistances based on the error in efferent flowrates and the control input, to obtain the
new x(t) vector. At each time step, inner iterations are used to find the equilibrium
solution for the combined system of equations at that time step. Each time step is
2
solved this way until the final time is reached. Hence, this model can be used, given
patient specific CoW geometry from MRI data, to predict the potential outcome of
surgical procedures or other therapies – a significant clinical tool.
Results
Control gains Kp, Ki, and Kd, are determined to match the clinical data from Newell et
al. (1994) by matching the percentage change and duration of the time-dependent
velocity profiles in the MCA. The complete CoW is modelled with venous pressures
of 4mmHg and an arterial pressure drop from 93mmHg to 73mmHg in the RICA
using the specific arterial resistances from Ferrandez et al. (2000). The time constant
 is taken to be 3 seconds, and the proportional, integral, and derivative gains are of
the order 0.1, 10, and 0.01, respectively. Simulations are carried out for the balanced
configuration and each case where a single circulus vessel is omitted. Where only
one vessel in the CoW is omitted, the system is able to obtain the reference efferent
flowrates. These results match the 20% drop in flowrate, and 20 second response
time of Newell et al. (1994) and the reductions in flowrate for the missing vessel
configurations are in good agreement with Ferrandez (2002), as shown in Table 1.
Table 1: Comparison of Results with Ferrandez
% drop in flow through RMCA after 20% pressure
drop in RICA
Balanced Configuration
Missing LPCA1
Missing LPCoA
Missing LACA1
Missing ACoA
Missing RACA1
Missing RPCoA
Missing RPCA1
(Ferrandez, 2002)
18
Not simulated
18
18
18
20
20
Not simulated
(Present Model)
19
19
19
20
20
21
19
19
A realistic high stroke risk case with decreases in radius of the LICA and RICA of
50% and 30% respectively, and a missing LPCA1 is modelled (Lodder et al., 1996).
This case represents a situation where the individual would be hypertensive to
maintain proper flow to the cerebral mass. When a pressure drop is simulated in the
RICA, LICA and BA, the LPCA2 cannot return to its reference flow value
representing a potential stroke, as shown in Figure 2.
Figure 2: Efferent Response to 20% Afferent Pressure Drop - High Stroke Risk Case
3
Discussion
The model created has simulated the omission of any single circulus artery and found
that no such omission leads to failure in reaching the reference (required) efferent
flowrates for a RICA stenosis. The omitted vessel simulations are in good agreement
with known physiological data and prior results, verifying the model dynamics. This
result highlights the known robustness of the CoW system in supplying the cerebral
mass territories. The high stroke risk case highlights the models’ ability to capture
more complex cases, and predict the response to changes in system dynamics.
Conclusions
The model is a significant improvement over pre-existing models in its incorporation
of inner iterations at each solution time-step, recognising the non-linearity of the
system, to find the equilibrium state within physiological limitations. Time varying
resistance accounts for vaso-dilation and vaso-constriction of the efferent arteries such
that a constant flowrate to the cerebral territories can be obtained after
occlusion/stenosis in any afferent artery. Results show good correlation with prior
results using CFD (Ferrandez et al., 2002) and the physiological data (Newell et al.,
1994). The solution for the CoW closed loop lumped parameter system is obtained in
a far shorter time period using this time-varying resistance model than with CFD
methods, and requires significantly less computational effort while retaining a high
level of accuracy. These results highlight the effectiveness of this computationally
efficient model as a tool for determining potential outcomes of surgical or other
therapies.
References
Alpers, B. and R. Berry (1963). "Circle of Willis in Cerebral Vascular Disorders, The
Anatomical Structure." Archives of Neurology 8: 398-402.
Ferrandez, A., T. David, et al. (2000). "Computational models of blood flow in the
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Ferrandez, A., T. David, et al. (2002). "Numerical models of auto-regulation and
blood flow in the cerebral circulation." Comput Methods Biomech Biomed
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Hillen, B., A. Drinkenburg, et al. (1988). "Analysis of Flow and Vascular Resistance
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Hillen, B., H. Hoogstraten, et al. (1986). "A Mathematical Model of the Flow in the
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Lodder, J., Hupperts, R., Boreas, A., Kessels, F. (1996). “The size of territorial brain
infarction on CT relates to the degree of internal carotid artery obstruction.”
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van der Zwan, A. and B. Hillen (1991). "Review of the variability of the territories of
the major cerebral arteries." Stroke 22(8): 1078-84.
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