The cerebral circulation is unique in its ability to maintain blood flow to the brain under widely varying physiologic conditions. Incorporating this autoregulatory response is necessary for cerebral blood flow (CBF) modeling, as well as investigations into pathological conditions. We discuss a one-dimensional (1D) nonlinear model of blood flow in the cerebral arteries coupled to autoregulatory lumped-parameter (LP) networks. The LP networks incorporate intracranial pressure (ICP), cerebrospinal fluid (CSF), and cortical collateral blood flow models. The overall model is used to evaluate changes in CBF due to occlusions in the middle cerebral artery (MCA) and common carotid artery (CCA). Velocity waveforms at the CCA and internal carotid artery (ICA) were examined prior and post MCA occlusion. Evident waveform changes due to the occlusion were observed, providing insight into cerebral vasospasm monitoring by morphological changes of the velocity or pressure waveforms. The role of modeling of collateral blood flows through cortical pathways and communicating arteries was also studied. When the MCA was occluded, the cortical collateral flow had an important compensatory role, whereas the communicating arteries in the circle of Willis (CoW) became more important when the CCA was occluded. To validate the model, simulations were conducted to reproduce a clinical test to assess dynamic autoregulatory function, and results demonstrated agreement with published measurements.

References

1.
Lassen
,
N. A.
,
1959
, “
Cerebral Blood Flow and Oxygen Consumption in Man
,”
Physiol. Rev.
,
39
(
2
), pp.
183
238
.
2.
Paulson
,
O. B.
,
Strandgaard
,
S.
, and
Edvinsson
,
L.
,
1989
, “
Cerebral Autoregulation
,”
Cerebrovasc. Brain Metab. Rev.
,
2
(
2
), pp.
161
192
.
3.
Van Beek
,
A. H.
,
Claassen
,
J. A.
,
Rikkert
,
M. G. O.
, and
Jansen
,
R. W.
,
2008
, “
Cerebral Autoregulation: An Overview of Current Concepts and Methodology With Special Focus on the Elderly
,”
J. Cereb. Blood Flow Metab.
,
28
(
6
), pp.
1071
1085
.
4.
Westerhof
,
N.
,
Lankhaar
,
J.-W.
, and
Westerhof
,
B. E.
,
2009
, “
The Arterial Windkessel
,”
Med. Biol. Eng. Comput.
,
47
(
2
), pp.
131
141
.
5.
Ursino
,
M.
, and
Giannessi
,
M.
,
2010
, “
A Model of Cerebrovascular Reactivity Including the Circle of Willis and Cortical Anastomoses
,”
Ann. Biomed. Eng.
,
38
(
3
), pp.
955
974
.
6.
Gonzalez-Fernandez
,
J. M.
, and
Ermentrout
,
B.
,
1994
, “
On the Origin and Dynamics of the Vasomotion of Small Arteries
,”
Math. Biosci.
,
119
(
2
), pp.
127
167
.
7.
Harder
,
D. R.
,
1984
, “
Pressure-Dependent Membrane Depolarization in Cat Middle Cerebral Artery
,”
Circ. Res.
,
55
(
2
), pp.
197
202
.
8.
Harder
,
D. R.
,
1987
, “
Pressure-Induced Myogenic Activation of Cat Cerebral Arteries is Dependent on Intact Endothelium
,”
Circ. Res.
,
60
(
1
), pp.
102
107
.
9.
David
,
T.
,
Alzaidi
,
S.
, and
Farr
,
H.
,
2009
, “
Coupled Autoregulation Models in the Cerebro-Vasculature
,”
J. Eng. Math.
,
64
(
4
), pp.
403
415
.
10.
van de Vosse
,
F. N.
, and
Stergiopulos
,
N.
,
2011
, “
Pulse Wave Propagation in the Arterial Tree
,”
Annu. Rev. Fluid Mech.
,
43
(
1
), pp.
467
499
.
11.
Alastruey
,
J.
,
Parker
,
K. H.
,
Peiró
,
J.
,
Byrd
,
S. M.
, and
Sherwin
,
S. J.
,
2007
, “
Modelling the Circle of Willis to Assess the Effects of Anatomical Variations and Occlusions on Cerebral Flows
,”
J. Biomech.
,
40
(
8
), pp.
1794
1805
.
12.
Alastruey
,
J.
,
Moore
,
S. M.
,
Parker
,
K. H.
,
David
,
T.
,
Peiró
,
J.
, and
Sherwin
,
S. J.
,
2008
, “
Reduced Modelling of Blood Flow in the Cerebral Circulation: Coupling 1-D, 0-D and Cerebral Auto-Regulation Models
,”
Int. J. Numer. Methods Fluids
,
56
(
8
), pp.
1061
1067
.
13.
Köppl
,
T.
,
Schneider
,
M.
,
Pohl
,
U.
, and
Wohlmuth
,
B.
,
2014
, “
The Influence of an Unilateral Carotid Artery Stenosis on Brain Oxygenation
,”
Med. Eng. Phys.
,
36
(
7
), pp.
905
914
.
14.
Liang
,
F.
,
Fukasaku
,
K.
,
Liu
,
H.
, and
Takagi
,
S. A.
,
2011
, “
Computational Model Study of the Influence of the Anatomy of the Circle of Willis on Cerebral Hyperperfusion Following Carotid Artery Surgery
,”
Biomed. Eng. Online
,
10
(
84
), pp.
1
22
.
15.
David
,
T.
, and
Moore
,
S.
,
2008
, “
Modeling Perfusion in the Cerebral Vasculature
,”
Med. Eng. Phys.
,
30
(
10
), pp.
1227
1245
.
16.
Connolly
,
M.
,
He
,
X.
,
Gonzalez
,
N.
,
Vespa
,
P.
,
DiStefano
,
J.
, III
, and
Hu
,
X.
,
2014
, “
Reproduction of Consistent Pulse-Waveform Changes Using a Computational Model of the Cerebral Circulatory System
,”
Med. Eng. Phys.
,
36
(
3
), pp.
354
363
.
17.
Lodi
,
C. A.
, and
Ursino
,
M.
,
1999
, “
Hemodynamic Effect of Cerebral Vasospasm in Humans: A Modeling Study
,”
Ann. Biomed. Eng.
,
27
(
2
), pp.
257
273
.
18.
Ursino
,
M.
, and
Lodi
,
C. A.
,
1998
, “
Interaction Among Autoregulation, CO2 Reactivity, and Intracranial Pressure: A Mathematical Model
,”
Am. J. Physiol.: Heart Circ. Physiol.
,
274
(
5
), pp.
H1715
H1728
.
19.
Giller
,
C. A.
,
1991
, “
A Bedside Test for Cerebral Autoregulation Using Transcranial Doppler Ultrasound
,”
Acta Neurochir.
,
108
(
1–2
), pp.
7
14
.
20.
Fahrig
,
R.
,
Nikolov
,
H.
,
Fox
,
A. J.
, and
Holdsworth
,
D. W.
,
1999
, “
A Three-Dimensional Cerebrovascular Flow Phantom
,”
Med. Phys.
,
26
(
8
), pp.
1589
1599
.
21.
Moore
,
S.
,
David
,
T.
,
Chase
,
J. G.
,
Arnold
,
J.
, and
Fink
,
J.
,
2006
, “
3D Models of Blood Flow in the Cerebral Vasculature
,”
J. Biomech.
,
39
(
8
), pp.
1454
1463
.
22.
Stergiopulos
,
N.
,
Young
,
D. F.
, and
Rogge
,
T. R.
,
1992
, “
Computer Simulation of Arterial Flow With Applications to Arterial and Aortic Stenoses
,”
J. Biomech.
,
25
(
12
), pp.
1477
1488
.
23.
Barnard
,
A. C. L.
,
Hunt
,
W. A.
,
Timlake
,
W. P.
, and
Varley
,
E. A.
,
1966
, “
Theory of Fluid Flow in Compliant Tubes
,”
Biophys. J.
,
6
(
6
), pp.
717
724
.
24.
Hughes
,
T. J. R.
, and
Lubliner
,
J.
,
1973
, “
On the One-Dimensional Theory of Blood Flow in the Large Vessels
,”
Math. Biosci.
,
18
(
1–2
), pp.
161
170
.
25.
Formaggia
,
L.
,
Lamponi
,
D.
, and
Quarteroni
,
A.
,
2003
, “
One-Dimensional Models for Blood Flow in Arteries
,”
J. Eng. Math.
,
47
(
3–4
), pp.
251
276
.
26.
Smith
,
N. P.
,
Pullan
,
A. J.
, and
Hunter
,
P. J.
,
2002
, “
An Anatomically Based Model of Transient Coronary Blood Flow in the Heart
,”
SIAM J. Appl. Math.
,
62
(
3
), pp.
990
1018
.
27.
Sherwin
,
S. J.
,
Franke
,
V.
,
Peiró
,
J.
, and
Parker
,
K.
,
2003
, “
One-Dimensional Modelling of a Vascular Network in Space-Time Variables
,”
J. Eng. Math.
,
47
(
3–4
), pp.
217
250
.
28.
Olufsen
,
M. S.
,
Peskin
,
C. S.
,
Kim
,
W. Y.
,
Pedersen
,
E. M.
,
Nadim
,
A.
, and
Larsen
,
J.
,
2000
, “
Numerical Simulation and Experimental Validation of Blood Flow in Arteries With Structured-Tree Outflow Conditions
,”
Ann. Biomed. Eng.
,
28
(
11
), pp.
1281
1299
.
29.
Steele
,
B. N.
,
Wan
,
J.
,
Ku
,
J. P.
,
Hughes
,
T. J.
, and
Taylor
,
C. A.
,
2003
, “
In Vivo Validation of a One-Dimensional Finite-Element Method for Predicting Blood Flow in Cardiovascular Bypass Grafts
,”
IEEE Trans. Biomed. Eng.
,
50
(
6
), pp.
649
656
.
30.
Parker
,
K. H.
,
2009
, “
An Introduction to Wave Intensity Analysis
,”
Med. Biol. Eng. Comput.
,
47
(
2
), pp.
175
188
.
31.
Alastruey
,
J.
,
Hunt
,
A. A. E.
, and
Weinberg
,
P. D.
,
2014
, “
Novel Wave Intensity Analysis of Arterial Pulse Wave Propagation Accounting for Peripheral Reflections
,”
Int. J. Numer. Methods Biomed. Eng.
,
30
(
2
), pp.
249
279
.
32.
Formaggia
,
L.
,
Lamponi
,
D.
,
Tuveri
,
M.
, and
Veneziani
,
A.
,
2006
, “
Numerical Modeling of 1D Arterial Networks Coupled With a Lumped Parameters Description of the Heart
,”
Comput. Methods Biomech. Biomed. Eng.
,
9
(
5
), pp.
273
288
.
33.
Alastruey
,
J.
,
Parker
,
K. H.
,
Peiró
,
J.
, and
Sherwin
,
S. J.
,
2008
, “
Lumped Parameter Outflow Models for 1-D Blood Flow Simulations: Effect on Pulse Waves and Parameter Estimation
,”
Commun. Comput. Phys.
,
4
(
2
), pp.
317
336
.
34.
Liang
,
F.
,
Takagi
,
S.
,
Himeno
,
R.
, and
Liu
,
H.
,
2009
, “
Multi-Scale Modeling of the Human Cardiovascular System With Applications to Aortic Valvular and Arterial Stenoses
,”
Med. Biol. Eng. Comput.
,
47
(
7
), pp.
743
755
.
35.
Huang
,
P.
, and
Muller
,
L.
,
2015
, “
Simulation of One-Dimensional Blood Flow in Networks of Human Vessels Using a Novel TVD Scheme
,”
Int. J. Numer. Methods Biomed. Eng.
,
31
(
5
), p.
e02701
.
36.
Hu
,
X.
,
Nenov
,
V.
,
Bergsneider
,
M.
,
Glenn
,
T.
,
Vespa
,
P.
, and
Martin
,
N.
,
2007
, “
Estimation of Hidden State Variables of the Intracranial System Using Constrained Nonlinear Kalman Filters
,”
IEEE Trans. Biomed. Eng.
,
54
(
4
), pp.
597
610
.
37.
Blacher
,
J.
,
Asmar
,
R.
,
Djane
,
S.
,
London
,
G. M.
, and
Safar
,
M. E.
,
1999
, “
Aortic Pulse Wave Velocity as a Marker of Cardiovascular Risk in Hypertensive Patients
,”
Hypertension
,
33
(
5
), pp.
1111
1117
.
38.
Sutton-Tyrrell
,
K.
,
Najjar
,
S. S.
,
Boudreau
,
R. M.
,
Venkitachalam
,
L.
,
Kupelian
,
V.
,
Simonsick
,
E. M.
,
Havlik
,
R.
,
Lakatta
,
E. G.
,
Spurgeon
,
H.
,
Kritchevsky
,
S.
,
Pahor
,
M.
,
Bauer
,
D.
, and
Newman
,
A.
,
2005
, “
Elevated Aortic Pulse Wave Velocity, A Marker of Arterial Stiffness, Predicts Cardiovascular Events in Well-Functioning Older Adults
,”
Circulation
,
111
(
25
), pp.
3384
3390
.
39.
Latham
,
R. D.
,
Westerhof
,
N.
,
Sipkema
,
P.
,
Rubal
,
B. J.
,
Reuderink
,
P.
, and
Murgo
,
J. P.
,
1985
, “
Regional Wave Travel and Reflections Along the Human Aorta: A Study With Six Simultaneous Micromanometric Pressures
,”
Circulation
,
72
(
6
), pp.
1257
1269
.
40.
Willemet
,
M.
, and
Alastruey
,
J.
,
2015
, “
Arterial Pressure and Flow Wave Analysis Using Time-Domain 1-D Hemodynamics
,”
Ann. Biomed. Eng.
,
43
(
1
), pp.
190
206
.
41.
Parker
,
K. H.
, and
Jones
,
C.
,
1990
, “
Forward and Backward Running Waves in the Arteries: Analysis Using the Method of Characteristics
,”
ASME J. Biomech. Eng.
,
112
(
3
), pp.
322
326
.
42.
Niki
,
K.
,
Sugawara
,
M.
,
Chang
,
D.
,
Harada
,
A.
,
Okada
,
T.
,
Sakai
,
R.
,
Uchida
,
K.
,
Tanaka
,
R.
, and
Mumford
,
C. E.
,
2002
, “
A New Noninvasive Measurement System for Wave Intensity: Evaluation of Carotid Arterial Wave Intensity and Reproducibility
,”
Heart Vessels
,
17
(
1
), pp.
12
21
.
43.
Asgari
,
S.
,
Gonzalez
,
N.
,
Subudhi
,
A. W.
,
Hamilton
,
R.
,
Vespa
,
P.
,
Bergsneider
,
M.
,
Roach
,
R. C.
, and
Hu
,
X.
,
2012
, “
Continuous Detection of Cerebral Vasodilatation and Vasoconstriction Using Intracranial Pulse Morphological Template Matching
,”
PLoS One
,
7
(
11
), p.
e50795
.
44.
Czosnyka
,
M.
,
Smielewski
,
P.
,
Kirkpatrick
,
P.
,
Menon
,
D. K.
, and
Pickard
,
J. D.
,
1996
, “
Monitoring of Cerebral Autoregulation in Head-Injured Patients
,”
Stroke
,
27
(
10
), pp.
1829
1834
.
45.
Miller
,
J. D.
, and
Becker
,
D. P.
,
1982
, “
Secondary Insults to the Injured Brain
,”
J. R. Coll. Surg. Edinburgh
,
27
(
5
), pp.
292
298
.
46.
Smielewski
,
P.
,
Czosnyka
,
M.
,
Kirkpatrick
,
P.
,
McEroy
,
H.
,
Rutkowska
,
H.
, and
Pickard
,
J. D.
,
1996
, “
Assessment of Cerebral Autoregulation Using Carotid Artery Compression
,”
Stroke
,
27
(
12
), pp.
2197
2208
.
47.
Liang
,
F.
,
Takagi
,
S.
,
Himeno
,
R.
, and
Liu
,
H.
,
2009
, “
Biomechanical Characterization of Ventricular–Arterial Coupling During Aging: A Multi-Scale Model Study
,”
J. Biomech.
,
42
(
6
), pp.
692
704
.
48.
Epstein
,
S.
,
Willemet
,
M.
,
Chowienczyk
,
P.
, and
Alastruey
,
J.
,
2015
, “
Reducing the Number of Parameters in 1D Arterial Blood Flow Modelling: Less is More for Patient-Specific Simulations
,”
Am. J. Physiol.: Heart Circ. Physiol.
,
309
(
1
), pp.
H222
H234
.
49.
Alastruey
,
J.
,
Khir
,
A. W.
,
Matthys
,
K. S.
,
Segers
,
P.
,
Sherwin
,
S. J.
,
Verdonck
,
P. R.
,
Parker
,
K. H.
, and
Peiró
,
J.
,
2011
, “
Pulse Wave Propagation in a Model Human Arterial Network: Assessment of 1-D Visco-Elastic Simulations Against In Vitro Measurements
,”
J. Biomech.
,
44
(
12
), pp.
2250
2258
.
50.
Reymond
,
P.
,
Merenda
,
F.
,
Perren
,
F.
,
Rüfenacht
,
D.
, and
Stergiopulos
,
N.
,
2009
, “
Validation of a One-Dimensional Model of the Systemic Arterial Tree
,”
Am. J. Physiol.: Heart Circ. Physiol.
,
297
(
1
), pp.
H208
H222
.
51.
Valdez-Jasso
,
D.
,
Bia
,
D.
,
Zócalo
,
Y.
,
Armentano
,
R. L.
,
Haider
,
M. A.
, and
Olufsen
,
M. S.
,
2011
, “
Linear and Nonlinear Viscoelastic Modeling of Aorta and Carotid Pressure–Area Dynamics Under In Vivo and Ex Vivo Conditions
,”
Ann. Biomed. Eng.
,
39
(
5
), pp.
1438
1456
.
52.
Coyle
,
P.
, and
Heistad
,
D.
,
1991
, “
Development of Collaterals in the Cerebral Circulation
,”
J. Vasc. Res.
,
28
(
1–3
), pp.
183
189
.
53.
Coyle
,
P.
, and
Heistad
,
D. D.
,
1987
, “
Blood Flow Through Cerebral Collateral Vessels One Month After Middle Cerebral Artery Occlusion
,”
Stroke
,
18
(
2
), pp.
407
411
.
54.
Alpers
,
B.
,
Berry
,
R.
, and
Paddison
,
R.
,
1959
, “
Anatomical Studies of the Circle of Willis in Normal Brain
,”
AMA Arch. Neurol. Psychiatry
,
81
(
4
), pp.
409
418
.
55.
Kahlert
,
P.
,
Al-Rashid
,
F.
,
Döttger
,
P.
,
Mori
,
K.
,
Plicht
,
B.
,
Wendt
,
D.
,
Bergmann
,
L.
,
Kottenberg
,
E.
,
Schlamann
,
M.
,
Mummel
,
P.
,
Holle
,
D.
,
Thielmann
,
M.
,
Jakob
,
H. G.
,
Konorza
,
T.
,
Heusch
,
G.
,
Erbel
,
R.
, and
Eggebrecht
,
H.
,
2012
, “
Cerebral Embolization During Transcatheter Aortic Valve Implantation: A Transcranial Doppler Study
,”
Circulation
,
126
(
10
), pp.
1245
1255
.
56.
Hu
,
X.
,
Xu
,
P.
,
Scalzo
,
F.
,
Vespa
,
P.
, and
Bergsneider
,
M.
,
2009
, “
Morphological Clustering and Analysis of Continuous Intracranial Pressure
,”
IEEE Trans. Biomed. Eng.
,
56
(
3
), pp.
696
705
.
57.
Hu
,
X.
,
Xu
,
P.
,
Asgari
,
S.
,
Vespa
,
P.
, and
Bergsneider
,
M.
,
2010
, “
Forecasting ICP Elevation Based on Prescient Changes of Intracranial Pressure Waveform Morphology
,”
IEEE Trans. Biomed. Eng.
,
57
(
5
), pp.
1070
1078
.
You do not currently have access to this content.