Biomedical Engineering Reference
In-Depth Information
Fig. 7.2.
Schema of the constrained mixture model of an arterial segment consisting of elastin,
multiple families of collagen fibres, and smooth muscle. Shown, too, are the
in vivo
configuration,
β
s
, which is taken as a reference
β
0
for the kinematics, a traction-free excised configuration,
β
1
,
and a radially-cut, nearly stress-free configuration
β
2
. The deformation gradients are denoted by
F
i
(
i
=
1
,
2
)
and
Φ
0
is the opening angle in
β
2
the enhanced activity of the fibroblasts, while the
media
may not change its thick-
ness or composition significantly [15]. Thus, a more detailed description of the wall
mechanics is necessary to capture transmural differences in biomechanical response.
Indeed the G&R model introduced in Sect. 7.2.3 can be easily applied to a thick-wall
formulation. Let us introduce, then, a thick wall model of the artery to investigate the
effects of transmural distributions of individual components and of their mechanical
properties on the axial retraction, opening angle, and overall stress field in the vessel.
It proves convenient to study salient aspects of arterial wall mechanics using the
semi-inverse approach of finite elasticity. In contrast to usual formulations, which
use the “stress-free” configuration as a reference, we use the current, stressed config-
uration as a computational reference. Two reasons motivate this choice. First, most
investigators prescribe the stress-free reference configuration (e.g., via an opening
angle) based on empirical observations and then seek consequences of this refer-
ence on the
in vivo
state of stress. In contrast, we seek to determine consequences of
material nonuniformity on both the
in vivo
state of stress and the residual stress re-
lated opening of a cut segment. Second, our work is motivated by the need to model
growth and remodelling processes mathematically, which necessarily occur in the
in
vivo
, stressed state. Hence, we prescribe the kinematics for an idealized axisymmet-
ric artery via two successive motions (see Fig. 7.2): mappings of material points from
a physiologically-relevant
in vivo
configuration
z
) associ-
ated with the finite extension and inflation of an intact cylindrical segment at time
s
to an intact but traction-free excised configuration
β
s
(with coordinates
r
,
θ
,
β
1
(with coordinates
ρ
,
ϑ
,
ζ
)and
then to a nearly stress-free, radially-cut configuration
β
2
(
R
,
Θ
,
Z
)
. These mappings
