Biomedical Engineering Reference
In-Depth Information
2.2.2 Tube law
Here we adopt a very simple tube law of the form
p
=
p
e
(
x
)+
ψ
(
A
;
K
)
,
(2.3)
where
ψ
(
A
;
K
)=
p
−
p
e
≡
p
trans
(2.4)
is the
transmural pressure
, the difference between the pressure in the vessel, the
internal pressure
, and the external pressure. Here we choose
A
A
0
m
1
ψ
(
A
;
K
)=
K
(
x
)
sign
(
m
)
−
,
(2.5)
with
√
π
E
(
x
)
h
0
(
x
)
A
0
(
K
(
x
)=
.
(2.6)
2
(
1
−
ν
)
x
)
Here
h
0
(
x
)
is the vessel thickness;
A
0
(
x
)
is the cross-sectional area of the vessel at
equilibrium,
p
trans
=
0;
E
(
x
)
is the Young's modulus,
ν
is the Poisson ratio and
m
is a real number different from zero.
The external pressure, assumed to be known, may be decomposed as follows
p
e
(
x
)=
p
atm
+
p
musc
(
x
)
,
(2.7)
where
p
atm
is the atmospheric pressure, assumed constant here, and
p
musc
(
x
)
is the
pressure exerted by the surrounding tissue. It is reasonable that
p
musc
(
x
)
be a function
of time as well. See [24] for a discussion on
p
musc
(
in the context of chronic venous
disease. For a fuller discussion on tube laws see, for example, [3, 4, 8, 17, 18].
x
)
2.3 Model for discontinuous properties
In this section we reformulate the mathematical model (2.1)-(2.6) so as to accommo-
date discontinuous variations of material properties such as Young's modulus and
wall thickness.
2.3.1 Equations
We consider a simple mathematical model consisting of the partial differential equa-
tions (2.1)-(2.2), along with the tube law (2.3)-(2.5) with
m
>
0. We assume wall
thickness
h
0
to be functions of axial distance
x
.How-
ever, the equilibrium cross-sectional area
A
0
is assumed to be constant, which for
practical applications is of course a serious limitation. For the purpose of this paper,
we consider such assumption to be adequate. A more general model comprising six
equations is currently being studied, which will include the variation of
A
0
. Then the
pressure gradient in (2.2) is
(
x
)
and Young's modulus
E
(
x
)
∂
x
p
=
ψ
A
∂
x
A
+
ψ
K
∂
x
K
+
∂
x
p
e
(
x
)
,
(2.8)
