Biomedical Engineering Reference
In-Depth Information
are the density and the viscosity, respectively, and
N cf , the interfaces in
the 3D model to be coupled with the 1D model of the arterial tree. This set of equa-
tions must be provided with proper coupling boundary conditions (naturally given by
the proposed non-classical extended variational formulation) and a proper constitu-
tive relation relating the displacement of the arterial wall with the pressure (structural
model). The coupling boundary conditions for
Γ i , i
=
1
,...,
γ =
1 are given by (see (9.51))
Q sa | 3 D , i =
u 3 D ·
n d
Γ i
i
=
1
,...,
N cf ,
(9.79)
Γ i
= P 3 D I
u 3 D ) n
P sa | 3 D , i n
2
με (
i
=
1
,...,
N cf ,
(9.80)
| Γ i
where the pairs
N cf , denote the restriction of the flow
rate and pressure in the systemic arteries which converge to the N cf coupling inter-
faces of the 3D model. As said in Sect. 9.4.1, here we observe the consequences of
having neglected the viscous effects due to axial gradients in the 1D model. On the
other hand, for the constitutive equation for the arterial wall, denoted by
(
Q sa | 3 D , i ,
P sa | 3 D , i )
, i
=
1
,...,
∂Ω w ,we
choose an independent rings wall model consistent to that used for the 1D model,
yielding
Eh o
R o δ 3 D +
kh o
R o ∂δ 3 D
P 3 D
P o =
t ,
(9.81)
u 3 D = ∂δ 3 D
n
=
v FR ,
(9.82)
t
where
∂Ω w in the direction of its
normal n . The displacement w FR of the frame of reference is extended to the interior
of the domain by considering its harmonic extension, obtained by solving a Laplace
problem in the reference fluid domain.
Other constitutive behaviours for the blood can be incorporated in the model,
giving rise to an equation similar to (9.76) but now valid for non-Newtonian fluids,
like the one obtained when the regularized Casson model is used. This fact does not
affect the coupling equations derived from Problem 3.
δ 3 D n
=
w FR is the displacement of the surface
9.5 Numerical applications
In this section we present two study cases (one open-loop, one closed-loop) to show
the capabilities of the heterogeneous approach for modelling the cardiovascular sys-
tem. These cases address two pathological situations in which aneurisms have de-
veloped in the abdominal aorta and in the cerebral artery, respectively. The blood
flow is modelled in both cases as a Newtonian fluid.
We approximate the problem using the finite-element method for both the 3D and
the 1D models. For the 3D problem the mini element is used with SUPG stabiliza-
tion, and a Crank-Nicolson finite-difference discretization in time is performed. In
particular, in view of our independent ring model for the structure, our fluid-structure
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