1D and 3D Models of Auto-Regulated
Cerebrovascular Flow
THE 26th ANNUAL INTERNATIONAL CONFERENCE OF THE IEEE
ENGINEERING IN MEDICINE AND BIOLOGY SOCIETY
K T Moorhead, S M Moore, J G Chase, T David, J Fink
Department of Mechanical Engineering
University of Canterbury
Christchurch, New Zealand
Structure of the CoW
Anterior (Front)
Anterior Cerebral Artery
A2 Segment
ACA2
Anterior Communicating Artery
ACoA
•
Responsible for
distributing blood to the
major regions of the brain
•
Blood can be re-routed
through the circle to
maintain homeostasis
•
Previous Models
Anterior Cerebral Artery
A1 Segment
ACA1
Middle Cerebral Artery
MCA
Internal Carotid Artery
ICA
Anterior Choroidal Artery
AChA
Posterior Communicating Artery
PCoA
Posterior Cerebral Artery
P1 Segment
PCA1
Posterior Cerebral Artery
P2 segment
PCA 2
Superior Cerebellar Artery
SCbA
Basilar Artery
BA
– No auto-regulation
Left Side
Right Side
Posterior (Back)
– No transient dynamics
1 D and 3 D Geometry
1 D Model
3 D Model
Porous Block
RRACA2
RLACA2
RLMCA2
RLAChA
RRACA1
RACoA
RLACA1
Anterior Communicating Artery
ACoA
RRMCA1
RLMCA1
Middle Cerebral Artery
MCA
RRMCA2
RLPCoA
RRPCoA
RRAChA
+ve
RRPCA2
RRPCA1
RLPCA2
Posterior Cerebral Artery
P2 segment
PCA2
Anterior Choroidal Artery
AChA
Posterior Cerebral Artery
P1 Segment
PCA1
Posterior Communicating Artery
PCoA
Basilar Artery
BA
RBA1
RLSCbA
Anterior Cerebral Artery
A1 Segmen t
ACA1
RRICA
RLICA
RLPCA1
Anterior Cerebral Artery
A2 Segment
ACA2
Superior Cerebellar Artery
SCbA
Internal Carotid Artery
ICA
RRSCbA
RBA2
Left Side
Right Side
•
Efferent arteries resistances time-variable
•
CAD reconstruction of MRA scan
•
Circulus and afferent artery resistances
constant
•
Porous block represents capillary bed effects
Dynamic Auto-Regulation
de PI feedback control law
u(t )  K p e  K i  edt  KStandard
d
dt
R  ( R  Rref )  u(t )
(1  0.95) R ref  R  (1  0.95) R ref
Control gains
match the time
dependent velocity
profile of the MCA
from thigh cuff
experiments of
Newell et al.
(1994) - 20 sec
response time for a
20% pressure drop
Resistance dynamics of contraction/dilation
Amount of change is limited
•
Resistance limits
• Deadband
• Memory
•
Peripheral resistance ratio
based on Hillen (1986)
6:3:4:75:75
•
Total influx = 12.5 cm3s-1
1 D Fluid Model
RRACA2
RLACA2
RLAChA
RRACA1
RACoA
RLACA1
RLMCA2
RRMCA1
RLMCA1
RRMCA2
Poiseuille Flow
RRICA
RLICA
RLPCoA
RRPCoA
RRAChA
+ve
RRPCA2
RLPCA1
R
RRPCA1
RLPCA2
RBA1
RLSCbA
8l
r 4
RRSCbA
RBA2
NO
Constant resistance between
nodes captured by simple
circuit analogy:
R
P2
P1
Error in flowrate
YES
q = qref?
Calculate new
flowrate
Change in
control input
Change in
resistance
q
P1  P2
q
R
System is highly nonlinear: A(x(t))*x(t) = b(t)
Solve system iteratively between resistance and
flow rates
3 D Model Geometry

 t dV   udA  0
V

V
u

dV  uu  dA   pIdA    dA   udV
t
k
V




Results – Ideal Configuration
Ipsilateral Efferent flowrates
•
All circulus vessels
present
•
20 mmHg pressure
drop in the RICA
•
Very good agreement
in efferent flux profiles
between models
Results – Ideal Configuration
Comparison of Flowrates for 1D and 3D Models - Balanced Configuration with Pressure Drop
Circulus Flowrates
6
1D Model Afferent and Circulus Flowrates
3D Model Afferent and Circulus Flowrates
5
•
1 D model ACoA experiences
greater pressure losses because
this artery is least well
approximated by Poiseuille
Flow
•
Increase resistance of the
ACoA 9-fold in the 1 D model
to produce same effective
resistance as 3 D model
1D Model Afferent and Circulus Flowrates
with Increased ACoA Resistance
Flowrate (cm^3/s)
4
3
2
1
0
BA2
LPCoA LPCA1
LICA
LACA1
ACoA RACA1
RICA
RPCoA RPCA1
-1
Vessel Name
Significant improvement
Results – Absent Ipsilateral ACA1
ACA2
Ipsilateral Efferent flowrates
•
1 D model has the
ACoA resistance
increased 9-fold as
previously
•
Ipsilateral ACA2 can
not reach its reference
flowrate even before a
pressure drop is
imposed
•
Good agreement
between models –
models get same
“wrong” answer
Conclusions
•
•
•
•
•
1 D and 3 D CoW models created
Models include non-linear dynamics of auto-regulation using PI
controller
Model verified against limited clinical data and prior research
Excellent agreement between models for efferent flux profiles
1 D ACoA not well approximated by Poiseuille flow 
increase ACoA resistance 9-fold to obtain good agreement in
circulus flowrates between models
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 and pathological
conditions
Punishment of the Innocent
Questions ???