Biomedical Engineering Reference
In-Depth Information
where
P
m
and
Q
m
are the mean values of transmural pressure and flow rate,
A
i
,
B
i
,
C
i
,and
D
i
are the Fourier coefficients, and
T
is the cardiac period. These functions
allow one to control arbitrarily the mean values, the wave form, the frequency of the
transmural pressure and blood flow rate, and also their relative phase as a function of
the G&R time variable
s
and therefore allow one to model separately the adaptation
of the artery to the perturbation of each of these features of the hemodynamic loads.
Consistently with membrane theory [20] as well as prior work on aneurysms [4],
vasospasm [21], and cerebral artery adaptation [37], the Cauchy membrane stress
components, on the cardiac cycle time scale, are computed as
s
,
t
+
1
λ
z
(
∂
W
∂λ
θ
act
θ
(
T
θ
(
s
,
t
)=
h
(
s
,
t
)
σ
s
,
t
)
,
(7.3)
s
,
t
)
s
,
t
+
1
λ
θ
(
∂
W
∂λ
z
act
z
T
z
(
s
,
t
)=
h
(
s
,
t
)
σ
(
s
,
t
)
,
(7.4)
s
,
t
)
ac
z
are the smooth
muscle active contributions to the Cauchy stress in the circumferential and ax-
ial directions, respectively, depending on the net constrictor concentration
C
(see
Sect. 7.2.3). The same relationships hold on the G&R time scale with all the quan-
tities depending only on
s
.
The mean values of pressure
P
m
(
act
θ
where
W
is the strain energy density of the wall tissue,
σ
and
σ
are assumed as reference
values of pressure and flow, respectively, at each G&R time instant
s
hence, using
Eqs. (7.3) and (7.4), the radial equilibrium on the G&R time scale is enforced by the
equation
s
)
and flow rate
Q
m
(
s
)
s
+
1
λ
z
(
W
∂λ
θ
∂
act
θ
(
h
(
s
)
σ
s
)=
P
m
(
s
)
a
(
s
)
.
(7.5)
s
)
On the cardiac cycle time scale we should, in principle, consider the inertial force;
thus the radial equilibrium would assume the form
s
,
t
+
1
λ
z
(
W
∂λ
θ
∂
2
act
h
(
s
,
t
)
σ
θ
(
s
,
t
)=
P
(
s
,
t
)
a
(
s
,
t
)
−
ρ
h
(
s
,
t
)
a
(
s
,
t
)
,
(7.6)
s
,
t
)
but inertial forces, as proved in [8] and [22], are largely negligible for the vast major-
ity of species and vessel types and the quasi-static version of the equilibrium equa-
tions can be assumed also on the cardiac cycle time scale.
7.2.2 Modelling transmural inhomogeneity
Thin wall models are useful to capture the overall mechanical behaviour of the ves-
sel and its evolution in response to perturbed hemodynamics, but cannot describe
the local transmural differences of the biomechanical response. As described in the
introduction, arterial layers are strongly inhomogeneous and populated by different
types of cells that respond in very different ways to the external stimuli. For exam-
ple, it is very common to observe adventitial hypertrophy in hypertension, due to
