Pulse wave propagation in large arteries:
Modelling, validation and applications
Jordi Alastruey1,2
Kim Parker1
Joaquim Peiro2
Spencer Sherwin2
Peter Weinberg1
Departments of Bioengineering1
and Aeronautics2
Annual and Final Repor
FORM AR
(For submission to: British Heart Foundation, R
Greater London House, 180 Hampstead Road
Email research@bhf.org.uk
Due on anniversary of start date of grant and within three months
Annu
1(Be
Junbrief,
2010
(For submission
be concise, use plain English wherever possible) Please
sub
Greater Londo
an electronic copy via email. (The form is available via email on reque
Pulse wave propagation
systole
left ventricle
Closed distribution network with an
incompressible fluid
diastole
Blood pressure
Arteries distend to accommodate the
sudden increase in volume caused by the
contraction of the left ventricle
Pulse wave propagation
(c~5-10 m/s, T~1 s, L~5-10 m)
1 Jun 2010
Annu
(For submission
Greater Londo
Modelling pulse wave propagation
Why?
• Morphology and functionality of the cardiovascular
system from pulse waveforms
• Valuable information for clinical diagnosis and
treatment of disease
How?
• 1-D reduced modelling
1 Jun 2010
Annu
(For submission
Greater Londo
The 1-D model
• Long-wave approximation
∂A ∂AU
+
=0
(p=p(x,t), U=U(x,t), A=A(x,t))
∂t
∂x
• Cross-sectional averages of U and p
∂U
∂U 1 ∂p
• Incompressible and Newtonian fluid
+U
+
= f (U , A)
∂t
∂x ρ ∂x
Elastic, homogeneous and thin arterial walls
• tube
ations (1) and (2) can
be
completed
with
the
law previously
1
1
β
• Constant arterial length
p = pext + ( A 2 − Ao2 )
A0
• Radius of curvature >> arterial radius
!√
#
"
β
4√
A − A0 ,
β(x) =
πhE,
(3)
A0
3
neous, incompressible and elastic arterialA(x,t)
wall with a thickness
p(x,t)
and a lumen area A0 (x) at the reference state (P, UE) = (0, 0).
h
teristics applied to Equations (1) to (3) (with f = 0 and β and
" propagated forwardx by
at changes in pressure and velocity !!are
ard $
by Wb at a speed U − c [24, 2], where Wf,b = U ± 4 (c − c0 )
)=
β
1/4
A
2ρA0
q in(t)
p in(t)
L
R
C
U(x,t)
is the pulse wave speed, c0 = c(Aq(x,t)
0 ), and U # c
l
heme with a Legendre polynomial spectral/hp spatial discretisaBashforth time-integration scheme is applied to solve Equations
This algorithm, which has been successfully
validated against
1 Jun 2010
Annu
(For submission
Greater Londo
The characteristic system
∂W f
∂W f
∂A ∂AU
+ (U + c )
=S
+
=0
∂t
∂x
∂t
∂x
∂Wb
∂Wb
∂U
∂U 1 ∂p
+ (U − c )
=S
+U
+
= f (U , A)
∂t
∂x
∂t
∂x ρ ∂x
β 1/ 4
ations (1) and (2) can
be
completed
with
the
tube
law
previously
c
=
A
W f ,b = U ± 4(c − c0 ),
1
1
β
2 ρA0
p = pext + ( A 2 − Ao2 )
Bernhard Riemann
(1826-1866)
A0
1  f ∂p dβ ∂p dA0 
!√
#
"

S =  −
−
β
4√
A − A0 ,
β(x) =
πhE,
ρ(3)
 A ∂β dx ∂A0 dx 
A0
3
neous, incompressible and elastic arterial wall with a thickness
and a lumen area A0 (x) at the reference state (P, U ) = (0, 0).
teristics applied to Equations (1) to (3) (with f = 0 and β and
Wf
at changes
p(x,t) in pressure and velocity are propagated forward by
ard $
by Wb at a speed U − c [24, 2], where Wf,b = U ± 4 (c − c0 )
)=
β
1/4
A
2ρA0
is the pulse wave speed, c0 = c(A0 ), and U # c
heme with a Legendre polynomial spectral/hp spatial discretisaU(x,t)
Bashforth time-integration scheme is applied to solve Equations
This algorithm, which has been successfully
validated against
1 Jun 2010
(x,t)
Wb(x,t)
Annu
(For submission
Greater Londo
Numerical solution
∂W f
∂W f
∂A ∂AU
+ (U + c )
=S
+
=0
∂t
∂x
∂t
∂x
∂Wb
∂Wb
∂U
∂U 1 ∂p
+ (U − c )
=S
+U
+
= f (U , A)
∂t
∂x
∂t
∂x ρ ∂x
β 1/ 4
ations (1) and (2) can
be
completed
with
the
tube
law
previously
c
=
A
W f ,b = U ± 4(c − c0 ),
1
1
β
2 ρA0
p = pext + ( A 2 − Ao2 )
Bernhard Riemann
(1826-1866)
A0
1  f ∂p dβ ∂p dA0 
!√
#
{
"

S =  −
−
β
4√
A − A0 ,
β(x) =
πhE,
ρ(3)
 A ∂β dx ∂A0 dx 
A0
3
scheme:
Discontinuous
Galerkin scheme
neous, Numerical
incompressible and
elastic arterial
wall with a thickness
spatial discretisation
and a lumen area A0 (x) at the reference state (P, U ) =• (0,Spectral/hp
0).
teristics applied to Equations (1) to (3) (with f = 0 and
and expansion
• βLegendre
L
R
order Adams-Bashforth time
• 2nd by
at changes inxpressure
and
x
e
evelocity are propagated forward
integration
ard $
by Wb at a speedΩUe − c [24, 2], where Wf,b = U ± 4 (c − c0 )
• Riemann solver based
β
1/4
) = 2ρA0 A
is the pulse wave speed, c0 = c(A0 ), and U # c
,
,
heme with a Legendre polynomial spectral/hp spatial discretisaSherwin, et al., J. Eng. Math. 2003
Bashforth time-integration scheme is applied to solve Equations
This algorithm, which has been successfully
validated against
1 Jun 2010
Annu
(For submission
Greater Londo
Boundary conditions
)!!
systole
/+,012-.
$!!
(!!
#!!
Systole
Diastole
left ventricle
'!!
Rµ
IN
Rµ
Q in
!
diastole
Q out
Q
Pin
!'!!
!
OUT
P out
!"#
!"$
*+,-.
!"% Pin
!"&
(b)
(a)
Qin
Rµ
Rµ
1
Pout
C '
2
Qout
Qin
Rµ
L
1
Rµ
Rµ
Q in
Qout
2
Q
Pin
PC
C
Pout
Pin
PC Pin
C
(c)
(a)
(d)
Qin
Pin
P out
Pout
Rµ
Rµ
1
2
Qout
Qin
R
Annu
1 Jun 2010
P
P
P
(For submission
Greater Londo
P
Bifurcations
The Author
March 22, 2007
(A2,U2)
Brief
(A
1,U1) Article
The Author
March 22, 2007
(A3,U3)
A1 U1 = A2 U2 + A3 U3
1
1
p(A1 ) + ρU12 = p(A2 ) + ρU22
2
2
1
1
p(A1 ) + ρU12 = p(A3 ) + ρU32
2
2
A1 U1 = A2 U2 + A3 U3
Wf (A1 , U1 )t = Wf (A1(1)
, U1 )t+∆t
1
1
p(A1 ) + ρU12 = p(A2 ) + ρU22
2
2
Wb (A2 , U2 )t = Wb (A2(2)
, U2 )t+∆t
1
1
p(A1 ) + ρU12 = p(A3 ) + ρU32
2
2
Wb (A3 , U3 )t = Wb (A3(3)
, U3 )t+∆t
1 Jun 2010
Annu
(For submission
Greater Londo
portant insight into cardiovascular physiology, especially if
h patient-specific data obtained from imaging techniques.
Test against in vitro data
l-defined experimental 1:1 replica of the 37 largest systemic arteries
7
test the8 numerical
solution of the non-linear, 1-D equations of
9
1
w wave propagation.
The parameters required by the numerical
6 in-vitro model and no data fitting is involved.
ctly 6measured
in
the
2
3
el
Numerical model
C
4
5
1. Pulsatile pump (left heart)
2. Catheter access
3. Two-leaflet aortic valve
4. Peripheral resistance tube
5. Flexible plastic tubing (veins)
6. Venous overflow
7. Venous return conduit
8. Buffering reservoir
9. Pulmonary veins
of silicone and blood is
water-glycerol mixture.
Each artery is modelled as a thin, elastic and
homogeneous tube, and blood as a Newtonian
and incompressible fluid. The 1-D equations are
solved in the time domain using a spectral/hp
discontinuous Galerkin scheme.
1-D FLOW EQUATIONS IN COMPLIANT VESSELS
"A "Au
#
!0
"t
"x
"u
"u 1 "p f
#u #
!
"t
"x & "x &
p ! p $ A ; x, t %
f ! f $A, u%
Conservation
of mass
Conservation of
momentum
Tube law
Viscous force
Matthys, Alastruey, Peiró, Khir,
Segers,Verdonck, Parker,
Sherwin. J. Biomech. 2007
1 Jun 2010
Annu
(For submission
Greater Londo
Aortic locations
14
12
10
0
0.2
0.4
t (s)
0.6
)
123 123
45. 45.
Q (ml/s)
-)*./0+,
'!!
!
12
0.2
0.4
t (s)
0.6
0.8
exp
num
100
0
!100
!'!!
)
!'!!
!
14
200
)
'!!
!
exp
num
10
0
0.8
#!!
#!!
-)*./0+,
16
exp
num
P (kPa)
P (kPa)
16
)
!
!"#
!"#
!"$
()*+,!"$
!"%
0
!"&
!"%
!"&
0.2
0.4
t (s)
0.6
0.8
()*+,
Matthys, Alastruey, Peiró, Khir,
Segers,Verdonck, Parker,
Sherwin. J. Biomech. 2007
1 Jun 2010
Annu
(For submission
Greater Londo
Peripheral locations
16
16
exp
num
exp
num
14
P (kPa)
P (kPa)
14
12
10
0
12
10
0.2
0.4
t (s)
0.6
0.8
0
5
5
exp
num
0.4
t (s)
0.6
3
2
0.8
exp
num
4
Q (ml/s)
Q (ml/s)
4
3
2
1
1
0
0
0.2
0.2
0.4
t (s)
★
0.6
0.8
0
0.2
0.4
t (s)
0.6
0.8
Average relative root-mean square errors < 4% in
pressure and < 19% in flow.
1 Jun 2010
Annu
(For submission
Greater Londo
Effect of nonlinearities
P (kPa)
16
∂A ∂AU
+
=0
∂t
∂x
∂U
∂U 1 ∂p
+U
+
= f (U , A)
∂t
∂x ρ ∂x
1
1
β
2
p = pext + ( A − Ao2 )
A0
14
12
10
0
★
β
3/2
2A0
0.2
0.4
t (s)
0.6
(A − A0 )
0.8
exp
num
lin
16
P (kPa)
p = pext +
exp
num
lin
14
12
10
0
0.2
0.4
t (s)
0.6
0.8
Average relative root-mean square errors < 4% in
pressure and < 19% in flow.
1 Jun 2010
Annu
(For submission
Greater Londo
enforced at junctions. Outflow terminal branches are
coupled to windkessel lumped parameter models
accounting for peripheral resistances and compliances [3].
Analysis of pulse waveforms
Total dy
(ascending aorta)
Terminal w
mod
4. Valve dynamics
1 Jun 2010
Overall,
dynamic
contribu
pressure
Annu
and valv
(For submission
Greater Londo
Analysis of pulse waveforms
How?
• Analysis of the peripheral, internal and valve dynamics
15
8
16
10
11
50
53
Peripheral
52
51 45
44
47
p (kPa)
10
24
25
1
2
t (s)
3
15
10
peripheral
conduit
internal &
5
55 54 48 49
internal &
conduit
valve
0
0
22
Internal
46
peripheral
5
p (kPa)
Valve
12
17
6
20
5
15
2 19
7 4 1 18
21
3 14 26
32 30
33
29 27
28 35
31
34
38
36
9 37 39
23
40
4143
42
13
0
9
valve
9.2
9.4 t (s) 9.6
9.8
10
Alastruey, Parker, Peiró, Sherwin. J. Eng. Math. 2009
1 Jun 2010
Annu
(For submission
Greater Londo
Peripheral dynamics
10
11
16
50
53
51 45
44
20
Ili
~
pw
Tho
17
22
24
25
p (kPa)
8
12
17
6
20
5
15
2 19
7 4 1 18
21
3 14 26
32 30
33
29 27
28 35
31
34
38
36
9 37 39
23
40
4143
42
13
14
Alastruey, Parker, Peiró,
Sherwin. J. Eng. Math. 2009
11
Asc
47
Bra
8
52
46
9
9.2
9.4
9.6 9.8
t (s)
10
10.2 10.4
55 54 48 49
1 Jun 2010
Annu
(For submission
Greater Londo
where pC (t) is the compliance-weighted space-average pressure of the arte
12
16
Peripheral
dynamics
network, C the total conduit compliance and q
c
C (t) =
$M
j
q
j=1 out the t
17
20
6
20 outflow to the periphery
driven by pC .
5
15
Ili
Tho
2 19
7 4 1 18
21 Hereafter it is assumed that ~Ri
17
pw0D = 0, i = 0, ..., N , since the resista
3 14 26
32 30
Alastruey, Parker, Peiró,
33
29 27
35 large arteries is much smaller than peripheral
Sherwin.
J. Eng. Math. 2009
28 in
resistances
[6][Chap 12
31
34
38
14
9 37 39 36
23 22
40
41
8
The works in [4, 5] have shown that during diastole Li0D → 0, i = 0, ...
43
42
24
% 11
10
50
44
25
i
i Asc→ ∞, i = 0, ..., N ; i.e, changes in pressure
51 45 so that c =
11
1/Li1D C1D
53
47
Bra
p (kPa)
13
8
flow occur synchronously.
Moreover, pi , i = 0, ..., N , converge to a sp
52
9
46
9.2
9.4
9.6 9.8
t (s)
10
independent pressure p!w that takes the form
55 54 48 49
p!w = Pout + (p!w (T0 ) − Pout )e
+
−t
R
e T CT
CT
&t
T0
'
qIN (t! ) +
T0 −t
RT CT
$M
M
#
1
1
=
j,
RT
j=1 Rj + Z0
1 Jun 2010
10.2 10.4
j
Cj Z0j Rj dqout
(t" )
j=1 R +Z j
dt"
j
0
CT = Cc + Cp ,
(
e
t"
RT CT
dt! ,
Cp =
M
#
j=1
11
t ≥ T0 ,
Rj Cj
Rj +
j,
Z0
Annu
(For submission
Greater Londo
Peripheral dynamics
10
11
16
50
53
51 45
44
20
Ili
~
pw
Tho
17
22
24
25
p (kPa)
8
12
17
6
20
5
15
2 19
7 4 1 18
21
3 14 26
32 30
33
29 27
28 35
31
34
38
36
9 37 39
23
40
4143
42
13
Alastruey, Parker, Peiró,
Sherwin. J. Eng. Math. 2009
14
11
Asc
47
Bra
8
52
46
9
9.2
9.4
TIME-DOMAIN WINDKESSEL
9.6 9.8
t (s)
10
10.2 10.4
H1363
55 54 48 49
Wang, O’Brien,
Shrive, Parker, Tyberg.
Am. J. Physiol. 2003
1 Jun 2010
Fig. 3. A: three-dimensional plot of PAo
versus time and distance (by 2-cm increments, from the aortic root to the
femoral artery). Data are from a single
dog. B: isobar contour plot of the same
(For submission
data (each line indicates an increment
Londo
of 2 mmHg). Note that, during Greater
late
diastole, pressure is dependent on time
Annu
Internal & valve dynamics
50
53
52
51 45
44
47
Pressure at the thoracic aorta
Dicrotic notch
22
Diastolic decay
15
24
25
p (kPa)
10
11
16
17
6
20
5
15
2 19
7 4 1 18
21
3 14 26
32 30
33
29 27
28 35
31
34
38
36
9 37 39
23
40
4143
42
13
8
12
46
10
peripheral
internal
proximal
5
0
9
55 54 48 49
& valve
9.2
9.4
t (s)
9.6
9.8
10
Alastruey, Parker, Peiró, Sherwin. J. Eng. Math. 2009
1 Jun 2010
Annu
(For submission
Greater Londo
Internal & valve dynamics
Delta function
50
53
51 45
44
Signal at the thoracic aorta
22
1.5
24
25
1
47
P
10
11
16
17
6
20
5
15
2 19
7 4 1 18
21
3 14 26
32 30
33
29 27
28 35
31
34
38
36
9 37 39
23
40
4143
42
13
8
12
1
0.8
52
46
0.5
P
0.6
0
0.4
0.2
0
55 54 48 49
0
0
0.2
t (s)
0.1
0.2
t (s)
0.3
0.4
0.5
0.4
Alastruey, Parker, Peiró, Sherwin. J. Eng. Math. 2009
1 Jun 2010
Annu
(For submission
Greater Londo
Internal & valve dynamics
Delta function
50
53
51 45
44
52
22
!"$
24
25
!"#
!
47
1
0.8
Signal at the thoracic aorta
-
10
11
16
17
6
20
5
15
2 19
7 4 1 18
21
3 14 26
32 30
33
29 27
28 35
31
34
38
36
9 37 39
23
40
4143
42
13
8
12
46
!!"#
P
0.6
0.4
!!"$
0.2
!
55 54 48 49
0
0
0.2
t (s)
0.4
!"#
!"$
()*+,
!"%
!"&
!"'
Alastruey, Parker, Peiró, Sherwin. J. Eng. Math. 2009
1 Jun 2010
Annu
(For submission
Greater Londo
Blood flow in the cerebral circulation
COUPLING 1-D, 0-D AND CEREBRAL AUT
ACA
ACoA
29
21
18
* 31 *25
26
24
*
MCA
30
20
13
I. Complet CoW
PCoA
28 27
22
*
33
12
23
*
19
10
*
32
11
V. ACA, A1 absent
PCA
14
17
6
5
II ACoA absent
www.health.allrefer.com
3
1
VI. PCA, P1 absent
4
2
9
U (m/s)
7
III. PCoA absent
16
15
8
IV. PCoAs absent
VII. PCoA &
contralateral
PCA, P1 absent
!
Metabolic auto-regulation model
Figure 1. Schematic representation of the 1-D arterial netwo
RCR windkessel
the anatomical variations
studied. model
The following arteries are
subclavians
(7,9),
brachials
Alastruey, Peiró, Parker, Byrd,
Sherwin.
J. Biomech.
2007(15,16), carotids (5,6,10,11,12,13,1
(19,20),
ACoA
(23,24),
Alastruey, Moore, Peiró, Parker, David,
Sherwin.
Int. J. (31),
Num. MCAs
Meth. Fluids
2007ACAs (25,26,29,30), and P
with RCR windkessel models (◦) or 0-D cerebral auto-regulat
Annu
velocity waveforms in ascending aorta (1), right common
ca
1 Jun 2010
(For submission
Greater Londo
1
1
p(A1 ) +2 ρU12 = p(A2 ) + 2ρU22
2
2
(2)
W (A , U1 ) = W (A
, U1)
(4)
Outflow
boundary
conditions
p(A ) + ρU = p(A ) + ρU
(3)
1 t
1
2
2
1
1
f
3
COUPLING 1-D, 0-D AND CEREBRAL AUT
1 t+∆t
2
2
3
ACA
Wb (A2 , UACoA
) = W (A , U )t+∆t
Wf (A1 , U21 )tt = Wfb (A12, U12)t+∆t
MCA
30
WW
(A23,,UU23)t+∆t
)t+∆t
3 ,2 U
b (A
, U32))tt =
=W
Wbb(A
b (A
13
(A ) − Pout
∗ 3 , ∗U3 )t P
WbA
(A
U = = Wb (A3 , U3 )t+∆t
∗
PCA
*
19
(5) 28(6)
27
*
* 22 32
33
12
(6) (7)
R
14
A U =
V. ACA, A1 absent
17
dt
A (t)−Ĉ
∗
=A U −
(13)
5
(8)
R
RL + RU e R
P (A∗ ) − PR(t)
Cµ =
R
dP (A∗A
) U =∗ ∗ RP1 (A∗ ) − Pout1 + eAR (t)−Ĉ
∗
10
11
dAOEF (t) ∗
6
P (A ∗=
) −GPOEF
out [CM RO2 − CBF (t) Ca CO2 OEF (t)],
∗ ∗
A∗ U∗ =
(7)
dt P (A ) − PC
II ACoA absent
R1
www.health.allrefer.com
C
23
20
I. Complet CoW
PCoA
(5)29
(4)26* 31 *25
24
*
21
18
R
,
,
7
Q in(8)
3
1
(9)
2
∗) − P
AOEF (t)−Ô
III.OEF
PCoA absent
P (ACBF(t)
dP (A∗ )
Q OEF
out L +
∗ ∗
Ue
C
= A U OEF
−
,
,(9)
ρc0 (t)R=
dt
+ eAOEF (t)−Ô 15 Pin (10)
Z0P=
1-D domain
P1out
in A0
ρc0
Z0 =
(10)
A0
dCt CO2 (t)
= CM RO2 − CBF (t)[Ct CO2 (t) −IV.
CaPCoAs
CO2 ],absent
dC
dtt CO2 (t)
(a)
Rµ
4
(11)
VI. PCA,
(14)P1 absent
Q out
9
C
8
16
(15)
PVII.
out PCoA &
U (m/s)
1
f
contralateral
PCA, P1 absent
!
(b)
(11)
dt
Metabolic auto-regulation model
Figure 1. Schematic representation of the 1-D arterial netwo
dAR (t)
RCRRwindkessel
= GR [Ct COR2SP
(12) Lmodel
R2µ(t)],
Rµ
µ1 − Ct CO
the
anatomical
variations
studied.
The following
are
µ
dA
(t)
Q
R
Qarteries
1
in
dt
2
2
Qout
out
=G
(12)
QRin[Ct CO2SP − Ct CO2 (t)],
subclavians
(7,9),
brachials
dt
Alastruey, Peiró, Parker, Byrd,
Sherwin.
J. Biomech.
2007(15,16), carotids (5,6,10,11,12,13,1
(19,20), ACoA (31), MCAs (23,24), ACAs (25,26,29,30), and P
Alastruey, Moore,
1 Peiró, Parker, David, Sherwin. Int. J. Num. Meth. Fluids 2007
with RCR windkessel models (◦) or 0-D cerebral auto-regulat
1
Annu
velocity
waveforms
in ascending
aorta (1), rightPcommon
ca
1 Jun 2010
(For submission
Pin
P
P
P
P
out
out Greater Londo
C
C
C
in
C
= CM RO2 − CBF (t)[Ct CO2 (t) − Ca CO2 ],
Inflow boundary condition
COUPLING 1-D, 0-D AND CEREBRAL AUT
ACA
ACoA
29
21
18
* 31 *25
26
24
*
MCA
30
20
13
I. Complet CoW
PCoA
28 27
22
*
33
12
23
*
19
10
*
32
11
V. ACA, A1 absent
PCA
14
17
6
5
II ACoA absent
www.health.allrefer.com
3
7
1
$!!
VI. PCA, P1 absent
4
2
9
U (m/s)
)!!
/+,012-.
III. PCoA absent
(!!
#!!
'!!
!
!'!!
!
Systole
Diastole
16
15
8
IV. PCoAs absent
VII. PCoA &
contralateral
PCA, P1 absent
!
Metabolic auto-regulation model
Figure 1. Schematic representation of the 1-D arterial netwo
RCR windkessel
the anatomical variations
studied. model
The following arteries are
!"#
!"$
!"%
!"&
'
*+,-.
subclavians
(7,9),
brachials
Alastruey, Peiró, Parker, Byrd,
Sherwin.
J. Biomech.
2007(15,16), carotids (5,6,10,11,12,13,1
(19,20),
ACoA
(23,24),
Alastruey, Moore, Peiró, Parker, David,
Sherwin.
Int. J. (31),
Num. MCAs
Meth. Fluids
2007ACAs (25,26,29,30), and P
with RCR windkessel models (◦) or 0-D cerebral auto-regulat
Annu
velocity waveforms in ascending aorta (1), right common
ca
1 Jun 2010
(For submission
Greater Londo
Carotid occlusion
COUPLING 1-D, 0-D AND CEREBRAL AUT
Effect of sudden carotid
occlusions on cerebral flows
30
29
21
18
* 31 *25
26
24
*
20
28 27
22
13
I. Complet CoW
*
33
12
23
*
19
10
*
32
11
V. ACA, A1 absent
14
17
6
5
II ACoA absent
3
Carotid endarterectomy
1
VI. PCA, P1 absent
4
2
9
U (m/s)
7
III. PCoA absent
Angioplasty
16
15
8
IV. PCoAs absent
Stenting
VII. PCoA &
contralateral
PCA, P1 absent
!
Metabolic auto-regulation model
Figure 1. Schematic representation of the 1-D arterial netwo
RCR windkessel
the anatomical variations
studied. model
The following arteries are
subclavians (7,9), brachials (15,16), carotids (5,6,10,11,12,13,1
ACoASherwin.
(31), MCAs
ACAs
and P
Alastruey, Moore, Peiró,(19,20),
Parker, David,
Int. J. (23,24),
Num. Meth.
Fluids(25,26,29,30),
2007
with RCR windkessel models (◦) or 0-D cerebral auto-regulat
Annu
velocity waveforms in ascending aorta (1), right common
ca
1 Jun 2010
(For submission
Greater Londo
Carotid occlusion
COUPLING 1-D, 0-D AND CEREBRAL AUT
30
29
21
18
* 31 *25
26
24
*
20
*
33
12
2
1
14
dAOEF (t)
=
G
[CM RO2 − CBF (t) Ca CO2 OEF
(t)],absent
II ACoA
12 OEF
dt
10
8
10
R(t) =
OEF (t)
RL + RU eAR (t)−Ĉ
1+
eAR (t)−Ĉ
19
10
*
32
11
V. ACA, A1 absent
14
6
17
5
(13)
7
3
2
(14)
1
,
*
VI. PCA, P1 absent
4
9
U (m/s)
16
P (kPa)
13
I. Complet CoW
28 27
22
23
III. PCoA absent
11OEF 12+ OEF
13 eAOEF
14 (t)−Ô15
L t (s)
U
=
,
1+
15
eAOEF (t)−Ô
1
1
p(A1 ) + ρU12 = p(A2 ) + ρU22 + Pdrop
2
2
(15)
16
8
IV. PCoAs absent
(16)
VII. PCoA &
contralateral
PCA, P1 absent
!
Metabolic auto-regulation model
Figure 1. Schematic representation of the 1-D arterial netwo
RCR windkessel
the anatomical variations
studied. model
The following arteries are
subclavians (7,9), brachials (15,16), carotids (5,6,10,11,12,13,1
ACoASherwin.
(31), MCAs
ACAs
and P
Alastruey, Moore, Peiró,(19,20),
Parker, David,
Int. J. (23,24),
Num. Meth.
Fluids(25,26,29,30),
2007
with RCR windkessel models (◦) or 0-D cerebral auto-regulat
Annu
velocity waveforms in ascending aorta (1), right common
ca
1 Jun 2010
(For submission
Greater Londo
Effect of carotid occlusion on MCA flow
COUPLING 1-D, 0-D AND CEREBRAL AUT
1.4
MCA
24
*
1
29
21
18
* 31 *25
26
20
13
I. Complet CoW
0.8
0.6
28 27
22
*
33
12
23
*
19
10
*
32
11
V. ACA, A1 absent
20 mmHg
20
25
30
t (s)
35
14
40
17
6
5
absent
Pdrop =II ACoA
20 mmHg
3
7
1
VI. PCA, P1 absent
4
2
9
U (m/s)
CBF
1.2
30
III. PCoA absent
16
15
8
IV. PCoAs absent
VII. PCoA &
contralateral
PCA, P1 absent
!
Metabolic auto-regulation model
Figure 1. Schematic representation of the 1-D arterial netwo
RCR windkessel
the anatomical variations
studied. model
The following arteries are
subclavians (7,9), brachials (15,16), carotids (5,6,10,11,12,13,1
ACoASherwin.
(31), MCAs
ACAs
and P
Alastruey, Moore, Peiró,(19,20),
Parker, David,
Int. J. (23,24),
Num. Meth.
Fluids(25,26,29,30),
2007
with RCR windkessel models (◦) or 0-D cerebral auto-regulat
Annu
velocity waveforms in ascending aorta (1), right common
ca
1 Jun 2010
(For submission
Greater Londo
Effect of carotid occlusion on MCA flow
COUPLING 1-D, 0-D AND CEREBRAL AUT
1.4
MCA
24
*
1
29
21
18
* 31 *25
26
20
13
I. Complet CoW
0.8
0.6
28 27
22
*
33
12
23
*
19
10
*
32
11
V. ACA, A1 absent
30 mmHg
20
25
30
t (s)
35
14
40
17
6
5
absent
Pdrop =II ACoA
30 mmHg
3
7
1
VI. PCA, P1 absent
4
2
9
U (m/s)
CBF
1.2
30
III. PCoA absent
16
15
8
IV. PCoAs absent
VII. PCoA &
contralateral
PCA, P1 absent
!
Metabolic auto-regulation model
Figure 1. Schematic representation of the 1-D arterial netwo
RCR windkessel
the anatomical variations
studied. model
The following arteries are
subclavians (7,9), brachials (15,16), carotids (5,6,10,11,12,13,1
ACoASherwin.
(31), MCAs
ACAs
and P
Alastruey, Moore, Peiró,(19,20),
Parker, David,
Int. J. (23,24),
Num. Meth.
Fluids(25,26,29,30),
2007
with RCR windkessel models (◦) or 0-D cerebral auto-regulat
Annu
velocity waveforms in ascending aorta (1), right common
ca
1 Jun 2010
(For submission
Greater Londo
Effect of carotid occlusion on MCA flow
COUPLING 1-D, 0-D AND CEREBRAL AUT
1.4
MCA
24
*
1
29
21
18
* 31 *25
26
20
13
I. Complet CoW
0.8
28 27
22
*
33
12
23
*
19
10
*
32
11
V. ACA, A1 absent
40 mmHg
0.6
20
25
30
t (s)
35
14
40
17
6
5
absent
Pdrop =II ACoA
30
mmHg
40
3
7
1
VI. PCA, P1 absent
4
2
9
U (m/s)
CBF
1.2
30
III. PCoA absent
16
15
8
IV. PCoAs absent
VII. PCoA &
contralateral
PCA, P1 absent
!
Metabolic auto-regulation model
Figure 1. Schematic representation of the 1-D arterial netwo
RCR windkessel
the anatomical variations
studied. model
The following arteries are
subclavians (7,9), brachials (15,16), carotids (5,6,10,11,12,13,1
ACoASherwin.
(31), MCAs
ACAs
and P
Alastruey, Moore, Peiró,(19,20),
Parker, David,
Int. J. (23,24),
Num. Meth.
Fluids(25,26,29,30),
2007
with RCR windkessel models (◦) or 0-D cerebral auto-regulat
Annu
velocity waveforms in ascending aorta (1), right common
ca
1 Jun 2010
(For submission
Greater Londo
Effect of carotid occlusion on MCA flow
COUPLING 1-D, 0-D AND CEREBRAL AUT
1.4
MCA
24
*
1
21
18
20
28 27
22
13
I. Complet CoW
0.8
*
33
12
20
25
30
t (s)
*
19
10
*
32
11
V. ACA, A1 absent
35
14
40
6
5
CtCO2
1.02
3
7
30 mmHg
1.01
1
20 mmHg
1
20
25
17
II ACoA absent
40 mmHg
1.03
0.99
1
23
40 mmHg
0.6
1.04
29
* 31 *25
26
2
9
III. PCoA absent
30
t (s)
35
40
16
15
0.95
8
IV. PCoAs absent
0.9
R
VI. PCA, P1 absent
4
U (m/s)
CBF
1.2
30
VII. PCoA &
contralateral
PCA, P1 absent
!
20 mmHg
0.85
0.8
40 mmHg
0.75
20
25
Metabolic auto-regulation model
Figure 1. Schematic representation of the 1-D arterial netwo
30 mmHg
RCR windkessel
the anatomical variations
studied. model
The following arteries are
subclavians (7,9), brachials (15,16), carotids (5,6,10,11,12,13,1
30
35
40
t (s)
ACoASherwin.
(31), MCAs
ACAs
and P
Alastruey, Moore, Peiró,(19,20),
Parker, David,
Int. J. (23,24),
Num. Meth.
Fluids(25,26,29,30),
2007
with RCR windkessel models (◦) or 0-D cerebral auto-regulat
Annu
velocity waveforms in ascending aorta (1), right common
ca
1 Jun 2010
(For submission
Greater Londo
Critical anatomical variations
COUPLING 1-D, 0-D AND CEREBRAL AUT
ACA
ACoA
30
* 31 *25
*
21
PCoA
28 27
22
13
*
33
12
23
*
18
20
I. Complet CoW
ACoA
26
24
MCA
29
19
10
*
32
11
V. ACA, A1 absent
PCA
14
17
6
5
II ACoA absent
www.health.allrefer.com
3
• Case V is the worst scenario in
terms of restoring normal
cerebral flows after a sudden
carotid occlusion.
• The ACoA is a critical collateral
1
VI. PCA, P1 absent
4
2
9
U (m/s)
7
III. PCoA absent
16
15
8
IV. PCoAs absent
VII. PCoA &
contralateral
PCA, P1 absent
!
pathway to compensate for
carotid occlusions.
Metabolic auto-regulation model
Figure 1. Schematic representation of the 1-D arterial netwo
RCR windkessel
model
the anatomical variations
studied. The
following arteries are
subclavians (7,9), brachials (15,16), carotids (5,6,10,11,12,13,1
ACoASherwin.
(31), MCAs
ACAs
and P
Alastruey, Moore, Peiró,(19,20),
Parker, David,
Int. J. (23,24),
Num. Meth.
Fluids(25,26,29,30),
2007
with RCR windkessel models (◦) or 0-D cerebral auto-regulat
Annu
velocity waveforms in ascending aorta (1), right common
ca
1 Jun 2010
(For submission
Greater Londo
Assessment of endothelium dysfunction
by wave analysis
!"#" $%&'(%)* !" #$
.),%8 9+38-("/ +*: *#$"#0 (;#:8 %2*$&8%#%
5<=
>?@A. #'$ (0& &L&,(* +8 O>P #'$ #,&(:6,0+6%'& /&"&
#-!9&'(&$ ;: A
Q#1"%'#*(. #' %'0%;%(+" +8 ,:,6%, R@P B
$&!"#$#(%+'3 C0& ,0#'!&* %' ;#*&6%'&
%'$-,&$ ;:
13 HIJ>
18
Measured by
8
/%(0 #6(&"#(%+'* +8
#,&(:6,0+6%'& +" <=>?@A /&"& #**+,%#(&$
17
/#2& ;&(/&&' 1&#D
photoplethysmography (0& *6+1& #'$ #916%(-$& +8 (0&161-6*&
9 12
*:*(+6& #'$ (0& $%,"+(%, '+(,0. ;-( (0&"& /#* '+ #6(&"#(%+'
19 +8
4
6
/#2&*1&&$. (0& %'(&"2#6*
,+91+'&'(* +8 (0&
11 ;&(/&&' $%L&"&'(
21
5 14
/#2&. +" (0& +2&"#66 #916%(-$& +8 (0& /#2&3
10 /#2&8+"9.
C0& *&,+'$#": 1&#D %' (0& 1-6*&
20 /0%,0 $&)'&*
2
(0& '+(,0. #"%*&* 8"+9 /#2& "&S&,(%+'*
(0& 2#*,-6#(-"&3 C
1 /%(0%'
15
7
>%,0+6* T NUH+-"D& VFWWXY 0#2& *0+/' (0#( (0& '#(-"& +8
3
(0& "&S&,(%+'* %'2+62&$ 2#"%&* #,,+"$%'! (+ ;+$: 16#'.
26
/#2&*1&&$. 0&#"( "#(& #'$ 9&#*-"%'!
*%(&3 N'& +;2%+-*
27 23
&B16#'#(%+' 8+" (0& 1"+9%'&'( $%#*(+6%,
1&#D
22%' (0& "#;;%(
38 353125
&#" %* (0#( %( %* # $%"&,( "&S&,(%+' +8 (0& *:*(+6%,24
/#2& 8"+9 (0&
28
6+/&" ;+$:. 1#"(%,-6#"6: 8"+939
(0& 6#"!&. 9-*,-6#" 0%'$6&!*3
29
30 6-91&$336+/&" ;+$:
C0& $%*(#',& 8"+9 (0& 0&#"( (+ (0&
"&S&,(%+' *%(& %* #11"+B%9#(&6: Z5 ,9
34%' (0&32"#;;%(. #'$ (0&
/#2&*1&&$ %* ;&(/&&' [3Z #'$ [3547
9 *!F V?2+6%+ !" #$3. FW\]Y3
37 36
C#D%'! # 9%$$6& 2#6-& 8+" (0& /#2&*1&&$.
#'$ %!'+"%'!
44 (0&
43
54
50
59 #'$ (0& &#" V*%',&
1#(0/#: ;&(/&&' (0& 0&#"(
%( /+-6$ #L&,(
40 49
51 &^-#66:Y !%2&* #531"&$%,(&$
(0& 1"%9#": #'$ "&S&,(&$
/#2&*
58
56
$&6#: ;&(/&&' (0& *:*(+6%, 1&#D 46
#'$ %(* 6+/&" 48
42 41 45;+$: "&S&,(%+'
+8 G3FZ * %' (0& &#"3 C0& $&6#: ;&(/&&' (0& *:*(+6%,
57 #'$
$%#*(+6%, 1&#D* 9&#*-"&$ %' (0& 1"&*&'( *(-$: /#*
G3FZ"G3GF * V9&#'"*3$3
8+" (0& FG #'%9#6*
55
52 +8 C"%#6 Z.
;&8+"& %'(&"2&'(%+'Y3 _&,#-*& +8 (0& &B,&66&'( #!"&&9&'(. /&
#**-9& (0%* 9&,0#'%*9 (+ ;& ,+""&,( %' (0& 8+66+/%'!
$%*,-**%+'3
(")%*+ , ?2&"#!& 1-6*& /#2&*3
C0"&&Harrington,
1-6*& /#2&* 8"+9
&#,0 +8Weinberg.
)2&
Nier,
Carrier,
Phys.
2008$%L&"&'( &L&,(* +8 2#*+$%6#(+"* ,+-6$
?(Exp.
6&#*(
(0"&&
#'%9#6* %'8-*&$ /%(0 #,&(:6,0+6%'& +" <=>?@A /&"& #2&"#!&$7 (0"&&
(0&+"&(%,#66: 6&#$ (+ # 8#66 %' (0& 0&%!0( +8 (0& '+(,0
/#2&* +;(#%'&$ 1"%+" (+ &#,0 %'8-*%+' /&"& #6*+ ,+9;%'&$3 C0&
Annu
#2&"#!& /#2&* /&"& *,#6&$ (+ !%2& (0& *#9& 1&#D #916%(-$&.
#'$ 2010
/&"&
1"&,&$%'! (0& $%#*(+6%, /#2&3 C0&*& #"& V%Y # $&,"&#*&
%'
1 Jun
(For submission
#6%!'&$ #( (0&%" 1&#D*3 C0&: /&"& ("-',#(&$ #( (0& 1+%'( /0&"& (0&
1-6*& /#2& 2&6+,%(:. V%%Y # "&$-,&$ $%#*(+6%, "&S&,(%+' +8Greater
(0&Londo
15
14
14
13
12
11
10
numerical
in vivo
13
12
5.5
5.6
5.7
10
5.8
5.5
5.6
5.7
0
5.5
5.8
5.6
5.7
5.8
100
50
0
5.7
11
Brachiocephalic
U (cm/s)
U (cm/s)
50
t (s)
12
10
5.8
50
0
5.5
5.6
t (s)
5.7
5.8
B
17
11
100
5.6
13
Descending Aorta
100
5.5
A
14
11
Ascending Aorta
U (cm/s)
15
P (kPa)
15
P (kPa)
P (kPa)
Model validation
5.5
5.6
t (s)
5.7
•Geometrical data measured from our arterial cast
•In vivo data from Avolio et al. Am. J. Phys. 1976
•Wave speeds from Milnor, Hemodynamics, 1989
5.8
8
13 18
16 9
4
5
10
2
1
12
19
21
6
14
20
7 15
3
26
27 23
38 353125 22
24 28
39
30 29 33
32
34
47 37
44
36
43
54
50
59
40 49
53
51
58
48
56
46 42
41 45
57
55
C
52
Alastruey, Nagel, Nier, Hunt, Weinberg, Peiró. J. Biomech. 2009
1 Jun 2010
Annu
(For submission
Greater Londo
Effect of model parameters on
auricular waveforms
Alastruey, Nagel, Nier, Hunt, Weinberg, Peiró. J. Biomech. 2009
1 Jun 2010
Annu
(For submission
Greater Londo
Conclusions
• Arterial pulse wave propagation can be simulated
using 1-D reduced modelling
• Tested against in vitro data
• Waveform analysis
• Application to clinically relevant problems
We have a tool to test hypothesis that cannot be
addressed in vivo for technical or physiological
reasons
Main challenge: Validation against in vivo data
1 Jun 2010
Annu
(For submission
Greater Londo