Biomedical Engineering Reference
In-Depth Information
Fig. 4
The characteristic
W 2
W 1
lines
0
L
A 4
A
c
( τ )
τ
β
1
4
0
W 1 , 2 =
u
±
d
τ =
u
±
4
A
.
(17)
2
ρ
A 0
A 0
Under physiological conditions, typical values of the flow velocity and mechan-
ical characteristics of the vessel wall are such that c
u , and consequently we
have that
0, everywhere. This means that the flow is subcritical,
such that the characteristic variable W 1 associated to the first eigenvalue,
λ 1 >
0and
λ 2 <
λ 1 ,travels
forward, while the characteristic variable W 2 , associated to the second eigenvalue,
λ 2 , travels backward (see Fig. 4 ). Hence, W 1 is the incoming characteristic, and W 2
is the outgoing characteristic, at the upstream left point ( z
=
a ), and vice versa at the
downstream right point ( z
b ), as illustrated in Fig. 4 .
Because of this, exactly one boundary condition must be imposed at each
extremity of the vessel [ 18 ]. However, the discretized model requires two conditions
at each boundary node in order to solve the system, corresponding to Q n + 1
h
=
and A n + 1
h
,
both at z
b . Thus two additional conditions, which have to be compatible
with the problem, are needed at the numerical level. These compatibility conditions
can be obtained by means of the outgoing characteristic at each boundary [ 18 ],
through projecting the equations along the characteristic lines exiting the domain
[ 16 ]. This results in computing the following additional relations at the boundaries:
=
a and z
=
Q h (
)
z a
Q n + 1
h
A n + 1
h
Q h (
A h (
W 2 (
(
a
) ,
(
a
)) =
W 2 (
z a ) ,
z a )) Δ
tK r
2 ,
at z
=
a
,
(18)
A h (
(
))
z a
and
Q h (
z b )
Q n + 1
h
A n + 1
h
Q h (
A h (
W 1 (
(
b
) ,
(
b
)) =
W 1 (
z b ) ,
z b )) Δ
tK r
2 ,
at z
=
b
,
(19)
A h (
(
z b ))
where z a and z b are the corresponding foot of the outgoing characteristic lines
which, using a first-order approximation [ 6 ], are given by
t Q h (
a
)
β
4
Q h (
A h (
A h (
z a =
a
Δ
t
λ 2 (
a
) ,
a
)) =
a
Δ
) +
A 0 (
a
))
,
(20)
A h (
a
2
ρ
t Q h (
b
)
β
4
Q h (
A h (
A h (
z b =
b
Δ
t
λ 1 (
b
) ,
b
)) =
b
Δ
)
A 0 (
b
))
.
(21)
A h (
b
2
ρ
 
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