Lumped Parameter and Feedback Control Lumped Parameter and Feedback Control Models of the Auto-Regulatory Response in Models of the Auto-Regulatory Response in

the Circle of Willisthe 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 EngineeringUniversity of Canterbury

Christchurch, New Zealand

Anterior

Posterior

Frontal lobe

Optic Chiasma

MCA

ICA

Pituitary Gland

Temporal lobe

Pons

Occipital lobe

ACA

ACoA

PCoA

PCA

BA

VA

Cerebellum

Structure of the CoWStructure of the CoW

• Geometry

• Purpose of CoW

• Auto-regulation

• > 50% do not have a complete CoW!

CFD model of the CoW

BA

LPCA1

LPCA2

LPCoA

LMCA

LACA1

LACA2

RACA2

ACoA

RACA1

RMCA

RPCoA RPCA2

RPCA1

LICA

+ve

+ve

RICA

BA

LPCA1

LPCA2

LPCoA

LMCA

LACA1

LACA2

RACA2

ACoA

RACA1

RMCA

RPCoA RPCA2

RPCA1

LICA

+ve

+ve

RICA

Research GoalsResearch 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 CoWModeling the CoW

R

P1P2

q

Constant resistance between

nodes captured by simplecircuit analogy:

Leads to system of linear equations for flowrates q(t) due to input conditions P(t):

Ax(t) = b(t)

RBA

RLPCA1

RLPCA2 RLPCoA

RLMCA

RLACA1

RLACA2

RRACA2

RACoA

RRACA1

RRMCA

RRPCoA RRPCA2

RRPCA1 RRICA

RLICA

+ve

Simplified geometry schematic of arterial system for basic dynamic analysis

4

8

r

lR

Poiseuille Flow

R

PPq 21

q Dynamic

Resistance R(t) Auto - Regulation

Controller

u e qref + -

Auto-Regulation ModelAuto-Regulation Model

vessel wall

q qref

smooth musclecells

Ca2+

1. Pressure/flow difference sensed 2. Ca2+ released into cytoplasm3. Muscle contraction4. Contracting/Dilating vessel radius5. Changing resistance of vessels

System is nonlinear:

A(x(t))*x(t) = b(t)

Error in flowrate

Change in control input

Change in resistance

Calculate new flowrate

q = qref?

NO

YES

refref RRR )95.01()95.01(

dt

deKedtKeKtu dip )(

)()( tuRRR ref Resistance dynamics of contraction/dilation

Standard PID feedback control law

Amount of change is limited

Simulations and Model ParametersSimulations 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 CasesResults – 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 before and after modelled stenosisafter modelled stenosis

•Flowrates normalised to LICA

•Red shows change in direction from steady state

-3

-2

-1

0

1

2

3

4

5

6

7

8

BA LPCA1 LPCoA LICA LACA1 ACoA RACA1 RICA RPCoA RPCA1 LPCA2 LMCA LACA2 RACA2 RMCA RPCA2

No

n-d

ime

ns

ion

al

Flo

wra

te

A

BA

LPCA 1

LPCA 2

LPCoA

LMCA

LACA 1

LACA 2

RACA 2

RACA 1

RMCA

RPCoA RPCA 2

RPCA 1

LICA

RICA

ACoA

Efferent Arteries-3

-2

-1

0

1

2

3

4

5

6

7

8

BA LPCA1 LPCoA LICA LACA1 ACoA RACA1 RICA RPCoA RPCA1 LPCA2 LMCA LACA2 RACA2 RMCA RPCA2

Before Stenosis After Stenosis/Occlusion in RICA

-3

-2

-1

0

1

2

3

4

5

6

7

8

BA LPCA1 LPCoA LICA LACA1 ACoA RACA1 RICA RPCoA RPCA1 LPCA2 LMCA LACA2 RACA2 RMCA RPCA2

No

n-d

ime

ns

ion

al

Flo

wra

te

BA

LPCA 1

LPCA 2

LPCoA

LMCA

LACA 1

LACA 2

RACA 2

RACA 1

RMCA

RPCoA RPCA 2

RPCA 1

LICA

RICA

ACoA

Missing ACoA case before and Missing ACoA case before and after modelled stenosisafter modelled stenosis

•Before stenosis, same flowrates as balanced case

•Red shows change in direction from steady state

•Efferent flowrates maintained

Note loss of communicating artery flow to support right side

-3

-2

-1

0

1

2

3

4

5

6

7

8

BA LPCA1 LPCoA LICA LACA1 ACoA RACA1 RICA RPCoA RPCA1 LPCA2 LMCA LACA2 RACA2 RMCA RPCA2

Before Stenosis After Stenosis/Occlusion in RICA

Efferent Arteries

Results – High Stroke Risk CaseResults – 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 CaseResults – 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

ConclusionsConclusions

• 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 InnocentPunishment of the Innocent

Questions ???