Biomedical Engineering Reference
In-Depth Information
2.3 Haemodynamics Modelling
2.3.1 Flow in a Single Vessel
In order to simulate the blood flow in an arterial tree, we must at first solve blood
flow equations in a single vessel. We assume that the flow in the circumferential
direction is negligible and that the radial velocity is small compared to axial
velocity, the governing Navier-Stokes equations can then be reduced to a 1D
formulation [
8
].
@
R
@t
þ V
@
R
R
2
@V
@x
þ
@x
¼
0
(1)
V
2
@
V
@t
þð
ÞV
@
V
R
@
R
1
r
@p
@x
¼
2
ua
a
V
R
2
;
2
a
1
@x
þ
2
ða
1
Þ
@x
þ
(2)
1
where (
1
) and (
2
) are the mass and momentum conservation equations, respec-
tively.
R
,
P
,
V
,
represent transmural vessel radius, blood pressure, velocity,
density and viscosity, respectively. The parameter
r
and
u
a
specifies axial velocity profile
V
x
, which is defined as: [
8
]
2
a
a
1
a
r
R
V
x
¼
V
1
(3)
2
a
The equation system (
1
)-(
2
) contains 3 unknowns (
P
,
V
and
R
) and is closed by
including a constitutive wall equation [
8
]:
R
0
1
Eh
0
R
0
R
2
2
3
P ¼
(4)
where
E
is Young's modulus of arterial wall,
h
0
,
R
o
are wall thickness and
unstressed radius, respectively. Equation (
4
) represents the relationship between
the transmural pressure and vessel radius.
Equations (
1
)-(
4
) form a nonlinear, hyperbolic system which cannot be solved
analytically. We adopted a second-order predictor-corrector type MacCormack
finite difference scheme to numerically solve these equations [
10
]. In this scheme,
the predicted values of
P
,
R
,
V
are evaluated by a backward difference method at
first, and their “corrected” values are evaluated by a forward finite difference. This
procedure is repeated at each time step. The stability requirement dictates that the
ratio of the numerical spatial and temporal step d
x
/d
t
must be faster than the wave
propagation velocity (which is about 10 m/s in large arteries). Hence, a temporal
step of 0.1 millisecond is required for a spatial step of 1 mm.
