Lumped Parameter and Feedback Control
Models of the Auto-Regulatory Response in
the Circle of Willis
World Congress on Medical Physics and Biomedical Engineering 2003
K T Moorhead, C V Doran, J G Chase, and T David
Department of Mechanical Engineering
University of Canterbury
Christchurch, New Zealand
Structure of the CoW
Anterior
Frontal lobe
ACA
Optic Chiasma
MCA
ACoA
ICA
PCoA
Pituitary Gland
PCA
Temporal lobe
CFD model of the CoW
BA
Pons
VA
Occipital lobe
Cerebellum
Posterior
RMCA
RPCoA
RPCA2
RACA1
+ve
RACA2
RPCA1
RICA
BA
ACoA
LICA
LACA2
+ve
LPCA1
LACA1
LPCoA
LMCA
LPCA2
•
•
•
•
Geometry
Purpose of CoW
Auto-regulation
> 50% do not have a
complete CoW!
Research Goals
• Desire: Better understand haemodynamics in the Circle of Willis (CoW)
cerebral arterial system
– Realistic dynamics for auto-regulation
– Match existing clinical data
• Goal: Create a simplified model of CoW haemodynamics to assist in
rapid diagnosis of stroke risk patients prior to surgery or other procedures
– Computationally simple
– Flexible
• Previous Work
– No auto-regulation (Hillen et al. 1988; Cassot et al. 2000)
– Steady state solution (Ursino and Lodi 1999; Hudetz et al. 1982)
In contrast, our model focuses on the transient dynamics
Modeling the CoW
Poiseuille Flow
R
RRMCA
8l
r 4
RRPCoA
RRACA1
RRACA2
Constant resistance between
nodes captured by simple
circuit analogy:
RRICA
RLICA
RBA
RLPCA1
RLACA1
RLPCoA
R
P2
P1
RRPCA1
+ve
RACoA
RLACA2
RRPCA2
RLPCA2
RLMCA
q
P  P2
q 1
R
Leads to system of linear equations for flow
rates q(t) due to input conditions P(t):
Ax(t) = b(t)
Simplified geometry schematic of arterial system
for basic dynamic analysis
Auto-Regulation Model
q qref
Ca2+
1.
2.
3.
4.
5.
vessel wall
smooth muscle
cells
u(t )  K p e  K i  edt  K d
de
dt
R  ( R  Rref )  u(t )
(1  0.95) R ref  R  (1  0.95) R ref
NO
Error in flowrate
YES
q = qref?
Calculate new
flowrate
Pressure/flow difference sensed
Ca2+ released into cytoplasm
Muscle contraction
Contracting/Dilating vessel radius
Changing resistance of vessels
Standard PID feedback control law
Resistance dynamics of contraction/dilation
Amount of change is limited
qref
+
e
-
Auto-Regulation
Controller
Change in
control input
Change in
resistance
System is nonlinear:
A(x(t))*x(t) = b(t)
u
Dynamic
Resistance R(t)
q
Simulations and Model Parameters
•
Reference and constant resistances based on known physiological data
•
Physiological data from thigh cuff experiments is used to determine control
gains
– Efferent resistances follow the ratio (6:3:4) for the ACA:MCA:PCA in the steady
state (Hillen et al., 1986)
– 20 sec response time for a 20% pressure drop (Newell et al., 1994 )
•
Drop in RICA of 20mmHg is tested to simulate a stenosis
•
Simulations run for a single vessel omission, testing each element of the CoW
– Verifies model against prior research using higher dimensional CFD methods
•
Simulation of a high risk stroke case with ICA blockage increasing resistance
– Illustrates potential of this model
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
Balanced
configuration before and
Balanced Configuration
after modelled stenosis
RMCA
•Flowrates normalised to LICA
RPCoA
RPCA2
RACA1
•Red shows change in direction from
steady state
RACA2
RPCA1
RICA
BA
ACoA
Balanced Configuration
LICA
LACA2
LPCA1
8
LACA1
7
LPCoA
LPCA2
6
Non-dimensional Flowrate
5
LMCA
A
4
3
2
LPCoA
LICA
LACA1
ACoA
RACA1
A
RICA
RPCoA RPCA1 LPCA2
LMCA
LACA2 RACA2
1
0
BA
LPCA1
LPCoA
LICA
LACA1
ACoA
RACA1
RICA
RPCoA RPCA1 LPCA2
LMCA
LACA2 RACA2
RMCA
-1
-2
Efferent Arteries
-3
Before Stenosis
After Stenosis/Occlusion in RICA
Before Stenosis
After
Stenosis/Occlusion in RICA
RPCA2
RMCA
RPCA2
Balanced Configuration
Missing ACoA case before and
after modelled stenosis
RMCA
•Before stenosis, same flowrates
as balanced case
RPCoA
•Red shows change in direction
from steady state
Missing ACoA
RACA2
RPCA1
RICA
BA
ACoA
•Efferent flowrates maintained
8
RPCA2
RACA1
LICA
LACA2
LPCA1
7
LACA1
LPCoA
6
A
5
Non-dimensional Flowrate
LPCA2
LMCA
4
3
2
LPCoA
LICA
LACA1
ACoA
RACA1
RICA
RPCoA RPCA1 LPCA2
LMCA
LACA2 RACA2
1
0
BA
LPCA1
LPCoA
LICA
LACA1
ACoA
RACA1
RICA
RPCoA RPCA1 LPCA2
LMCA
LACA2 RACA2
RMCA
RPCA2
-1
Efferent Arteries
-2
-3
Before Stenosis
Before Stenosis
After Stenosis/Occlusion in RICA
After Stenosis/Occlusion in RICA
Note loss of communicating artery flow to support right side
RMCA
RPCA2
Results – High Stroke Risk Case
• High stroke risk case:
– LICA and RICA radii reduced 50% and 40% respectively, representing
potential carotid artery blockages
– LPCA1 (Left Proximal Posterior Cerebral Artery) is omitted
– 20mmHg pressure drop in RICA simulating a stenosis is simulated
• This individual would be hypertensive to maintain steady state flow
requirements – captured by model.
– 93mmHg does not maintain reference flow rates in several efferent
arteries, even at maximum dilation
– ~113mmHg required to attain desired level.
Case is not common in all individuals but is encountered in
those needing an endarterectomy
Results – High Stroke Risk Case
LEFT
RIGHT
LPCA 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 non-linear dynamics of auto-regulation
Iterative solution method developed enabling rapid diagnosis
Model verified against limited clinical data and prior research
Several simulations 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
Punishment of the Innocent
Questions ???