Biomedical Engineering Reference
In-Depth Information
and the model of the cardiac valves are based on the data reported in [22, 24, 40].
The mechanical parameters of arterial walls in the 3D models match those of the 1D
model to which the specific vessel is coupled. As for the values of the blood density
and viscosity, we take
04 g/cm
3
04 dyn s/cm
2
, respectively.
ρ
=
1
.
and
μ
=
0
.
9.4.3 1D models
The arterial blood flow is modelled as the flow of a fluid in 1D compliant vessels in
order to capture the wave propagation phenomena. The governing equations for the
1D portion of the arterial system (all arterial segments) are derived from the Navier-
Stokes equations by introducing suitable geometrical and kinematical assumptions
(see (9.48)). This procedure yields the following set of partial differential equations
(see [25] for details):
Q
sa
,
m
A
sa
,
m
∂
Q
sa
,
m
∂
+
∂
∂
A
sa
,
m
ρ
∂
P
sa
,
m
∂
x
−
π
D
ρ
τ
=
−
=
,...,
,
β
o
sa
,
m
m
1
N
sa
(9.54)
t
x
∂
A
sa
,
m
∂
+
∂
Q
sa
,
m
∂
=
0
m
=
1
,...,
N
sa
,
(9.55)
t
x
with
|
|
f
r
ρ
u
sa
,
m
u
sa
,
m
τ
o
sa
,
m
=
Q
sa
,
m
=
u
sa
,
m
A
sa
,
m
,
(9.56)
8
where
Q
sa
,
m
is the flow rate,
A
sa
,
m
is the cross-sectional area of the artery (
D
its di-
ameter),
u
sa
,
m
the mean value of the axial velocity,
x
the axial coordinate,
P
sa
,
m
the
mean pressure,
τ
osa
,
m
the viscous shear stress acting on the arte-
rial wall,
f
r
a Darcy friction factor (in this work a fully developed parabolic velocity
profile is considered) and
ρ
the blood density,
β
is the momentum correction factor (
β
=
1 is considered
here).
The system is closed by introducing a constitutive law for the arterial wall. Here
the following visco-elastic model is used [29, 30]
A
sa
,
m
A
o
−
1
Eh
o
R
o
P
sa
,
m
=
P
o
+
Kh
o
R
o
2
A
o
A
sa
,
m
∂
1
A
sa
,
m
∂
+
m
=
1
,...,
N
sa
,
(9.57)
t
R
being the radius of the artery,
E
an effective Young modulus,
K
the viscosity of
the wall,
h
the thickness of the arterial wall. The subscript 'o' denotes quantities
evaluated at the reference pressure
P
o
.
The arterial tree is described by 128 arterial segments (see [1]) in which continu-
ity of mass and continuity of pressure are imposed at arterial junctions. Calling
N
jn
the total number of junctions with
N
cg
,
j
arterial segments converging to junction
j
