Biomedical Engineering Reference
In-Depth Information
Table 9.7.
Data used in the non-ideal diode models of the cardiac valves
Valves
Tricuspid
Pulmonar
Mitral
Aortic
R
[dyn cm
−
2
sml
−
1
]
0.006
0.006
0.006
0.006
L
[dyn cm
−
2
s
2
ml
−
1
]
0.005
0.005
0.005
0.005
B
[dyn cm
−
2
s
2
ml
−
2
]
0.0064
0.00756
0.0064
0.00756
θ
max
[
◦
]
75.0
75.0
75.0
75.0
θ
min
[
◦
]
5.0
5.0
5.0
5.0
I
[rad s
−
2
dyn
−
1
cm
2
]
k
P
/
4.126032
4.126032
4.126032
4.126032
I
[s
−
1
]
k
F
/
50.0
50.0
50.0
50.0
I
[rad s
−
1
ml
−
1
]
k
Q
/
2.0
2.0
2.0
2.0
I
[rad s
−
1
ml
−
1
]
k
V
/
3.5
3.5
3.5
7.0
where
θ
max
is the maximum angle of opening. A heuristic angular momentum bal-
ance equation for
θ
is used here (see [31] for the derivation and details) as follows:
k
V
Q
o
sin
I
d
2
θ
dt
2
k
F
d
dt
=
(
2
θ
)
if
P
i
≥
P
o
+
k
P
(
P
i
−
P
o
)+
k
Q
Q
o
cos
θ
+
P
o
,
(9.74)
0
if
P
i
<
where
I
is the momentum of inertia of the valve. The solution of this balance equation
is constrained to
θ
min
if
θ
<
θ
min
,
θ
=
(9.75)
θ
max
if
θ
>
θ
max
.
In this way, a valve can undergo malfunctioning in two distinct ways (or combina-
tion of them): by
stenosis
, if the valve is narrowed (reduced value of
θ
max
), and by
incompetence
or
insufficiency
, when the valve is leaky and fails to prevent promi-
nent backward flow (increased value of
θ
min
). The data used in the computational
implementation is presented in Table 9.7
9.4.8 3D models
As already stated, we consider the existence of 3D models accounting for all the com-
plexity of three-dimensional blood flow in specific vessels of interest. Then we make
use of the Navier-Stokes equations in moving domains (ALE formulation), that is
ρ
∂
u
3
D
∂
+
ρ∇
u
3
D
(
u
3
D
−
v
FR
)
−
μ
u
3
D
+
∇
P
3
D
=
f
in
Ω
,
(9.76)
t
=
Ω
,
div
u
3
D
0
in
(9.77)
plus proper coupling conditions (see Sect. 9.3) at
Γ
i
=
1
,...,
N
cf
,
(9.78)
i
where
u
3
D
is the fluid velocity,
v
FR
the velocity of the frame of reference consistent
with the ALE formulation,
P
3
D
the pressure field,
f
the volume body force,
ρ
and
μ
