Biomedical Engineering Reference
In-Depth Information
2.2 Mathematical models
Consider the geometric situation described in Fig. 2.1, which depicts a model for
a blood vessel. The mathematical model will assume one-dimensional flow in the
axial direction
x
.
2.2.1 Review of the basic equations
The basic equations for the flow of blood in medium-size to large arteries and veins
are obtained from the principles of conservation of mass
∂
t
A
+
∂
x
(
uA
)=
0
(2.1)
and momentum
A
ρ
∂
x
p
Au
2
ˆ
∂
t
(
uA
)+
∂
x
(
α
)+
=
−
Ru
.
(2.2)
A
(
x
,
t
)
is the cross-sectional area of the vessel or tube at position
x
and time
t
,
u
(
x
,
t
)
is the averaged velocity of blood at a cross section,
p
(
x
,
t
)
is pressure,
ρ
is the density
of blood, assumed constant, and
R
0 is the viscous resistance of the flow per unit
length of the tube, assumed to be known. We assume ˆ
>
1 in the momentum equa-
tion (2.2). There are two governing partial differential equations, (2.1), (2.2), and
three unknowns, namely
A
α
=
. An extra relation is required to
close the system. This is provided by the
tube law
, which relates the pressure
p
(
x
,
t
)
,
u
(
x
,
t
)
and
p
(
x
,
t
)
(
x
,
t
)
to the wall displacement via the cross-sectional area
A
. The tube law couples the
elastic properties of the vessel to the fluid dynamics and is analogous to the
equation
of state
in gas dynamics [22].
(
x
,
t
)
Fig. 2.1.
Assumed axially symmetric vessel configuration in three space dimensions at time t. Cross
sectional area
A
(
x
,
t
)
and wall thickness
h
0
(
x
)
are illustrated
