Cerebral Haemodynamics
and Auto-Regulatory Models of the
Circle of Willis
K T Moorhead, C V Doran, J G Chase, and T David
University of Canterbury
Dept of Mechanical Engineering
Research Goals
• Desire: Better understand cerebral haemodynamics in the Circle of Willis
cerebral arterial system (CoW)
– Including realistic dynamics for auto-regulation phenomenon
– Account for “essential dynamics” of the system
– Match existing clinical data and/or understanding
• Goal: Create a simplified model of CoW haemodynamics to assist in
rapid diagnosis of stroke risk patients prior to surgery or other procedures
– Must be computationally simpler than higher dimensional CFD
– Must be flexible enough to model different geometry’s and conditions of the
CoW arterial system
Circle of Willis (CoW) Definitions
Abbreviation
BA
LPCA1
LPCoA
LICA
LACA1
ACoA
RACA1
RICA
RPCoA
RPCA1
Vessel Full Name
Basilar Artery
Left proximal Posterior Cerebral Artery
Left Posterior Communicating Artery
Left Internal Carotid Artery
Left proximal Anterior Cerebral Artery
Anterior Communicating Artery
Right proximal Anterior Cerebral Artery
Right Internal Carotid Artery
Right Posterior Communicating Artery
Right proximal Posterior Cerebral Artery
LPCA2
LMCA
LACA2
Left distal Posterior Cerebral Artery
Left Middle Cerebral Artery
Left distal Anterior Cerebral Artery
RACA2
RMCA
RPCA2
Right distal Anterior Cerebral Artery
Right Middle Cerebral Artery
Right distal Posterior Cerebral Artery
Simplified geometry schematic of arterial system for basic dynamic analysis
Modeling the CoW
R
P2
P1
q
Constant resistance between
nodes captured by simple
circuit analogy:
Q = (P1 – P2)/R
Leads to system of linear equations for flow
rates q(t) due to input conditions p(t):
Ax(t) = b(t)
Auto-regulation not included yet
Auto-Regulation
• Dynamic phenomena where in blood perfusion pressure to efferent
arteries (L and R: ACA, MCA, PCA) is maintained within set limits.
• Feedback controlled system based on the body’s sensing of changes of
flow rate from a set (desired) reference.
• Control input “u” changes the dilation/contraction of the arterioles
within the cerebral territory supplied by the efferent artery to increase
(flow to high) or decrease (flow to low) resistance.
Auto-regulation modeled as feedback controlled, time varying resistances
on efferent arteries, creating a non-linear system
Auto-Regulation Model
q qref
vessel wall
Ca
smooth muscle
cells
R  ( R  Rref )  u(t )
u(t )  K p e  K i  edt  K d
de
dt
(1  0.95) R ref  R  (1  0.95) R ref
Resistance is a function of error,e(t)
Error is a function of flow rates
Flow rates are what you solve for
1. Pressure and/or flow that is different than
desired is sensed
2. Ca ions released
3. Triggering muscle fiber
contraction/expansion
4. Contracting/Dilating vessel radius
5. Changing resistance of vessels
Resistance dynamics of contraction/dilation
Standard PID feedback control law
Amount of change is limited
System is nonlinear: A(x(t))*x(t) = b(t)
Solve it iteratively between resistance
and flow rates
Model Parameters
• Reference and constant resistances based known physiological data
– Lengths, radii, etc
– Communicating arteries generally have very high resistance
• Geometry may commonly have one omitted element
• Physiological data from thigh cuff experiments is used to determine
control gains
– Correlated to match 20% drop and 20 second return time in clinical data
for 20 mmHg drop in afferent blood pressure
– Reference resistances for efferent arteries determined to match
physiological data which states that efferent resistances follow the ratio
(6:3:4) for the ACA:MCA:PCA arteries in the steady state
• Boundary conditions:
– Afferent pressure: 93mm Hg
– Efferent pressure: 4mm Hg
Simulations Run
• Drop in RICA of 20mm Hg is tested to simulate a stenosis
• Simulations run for a single vessel omission, testing each element of
the CoW
– Used to verify model versus prior research using higher dimensional (2D)
CFD methods that are computationally intense
• Simulation of a high risk stroke case with ICA blockage increasing
resistance
– To illustrate potential of this model.
• All simulations run for symmetric configuration where resistances on
L and R sides are not varied.
Results – Omitted Artery Cases
% drop in flow through RMCA after 20%
pressure drop in RICA
(Ferrandez, 2002)
(Present Model)
Balanced
Configuration
18
19
Missing LPCA1
Not simulated
19
Missing LPCoA
18
19
Missing LACA1
18
20
Missing ACoA
18
20
Missing RACA1
20
21
Missing RPCoA
20
19
Missing RPCA1
Not simulated
19
•No failure to return to qref flow
•Return times ~15-25 seconds
•Shows robustness of CoW system in
maintaining flow and pressure
Results – Omitted Artery Cases
Change in flow rate due
to pressure drop:
No omitted arteries
Red shows change direction from steady state
ACoA omitted – common anomaly
All results are % change of steady state values
Prior to pressure drop
Note loss of communicating artery flow to support right side
Results – High Stroke Risk Case
• High stroke risk case:
– LICA and RICA radii reduced 50% and 30% respectively, representing
potential carotid artery blockages
– LPCA1 (Left Proximal Posterior Cerebral Artery) is omitted
– 20mm Hg pressure drop in RICA simulating a stenosis is simulated
• This individual would be hypertensive to maintain steady state flow
requirements – captured by model.
– 93mm Hg does not maintain reference flow rates in several efferent
arteries, even at maximum dilation
– ~113mm Hg required to attain desired level.
Case is not common in all individuals but is encountered in
those needing carotid artery surgery
Results – High Stroke Risk Case
LEFT
RIGHT
LMCA fails to achieve desired flow rate indicating a potential stroke risk
under any procedure which entails such a pressure drop
Conclusions
• A new, simple model of cerebral haemodynamics created
• Model includes dynamics of auto-regulation and is non-linear
• Iterative solution method developed offering fast solution and enabling
use as a rapid diagnosis “what-if” tool.
• Model verified against limited clinical data and prior research using
higher dimensional CFD
• Several simulations run for omitted artery cases illustrate the
robustness of the CoW
• High stroke risk case illustrates the potential for simulating patient
specific geometry and situation to determine risk
• Future work includes more physiologically accurate auto-regulation
and geometry modelling, more clinical verification using existing data,
and modelling of greater variety of potential structures (eg fused
vessels).
Punishment of the Innocent
Questions ???