Multi-Scale Modelling of Vascular Disease: Abdominal Aortic Aneurysm Evolution - Computational Modeling in Tissue Engineering - page 314

protrusion; the posterior region is often constrained from radial expansion by the
adjacent spinal column [ 38 ]. To simulate this, we follow Watton et al. [ 15 ] and
model the spine as a stiff spring-backed plate [ 15 ]. Where the aneurysm wall
penetrates the plate, a penalty pressure acts normal to the aneurysm. The effective
pressure acting on the membrane, is given by the difference of internal physio-
logical pressure and the penalty pressure. For further details, see [ 43 ]. Two models
of AAA evolution are illustrated: case (i) prescribed elastin degradation; case (ii)
the degradation of elastin is driven by low WSS. In both examples, as the
geometry evolves, the collagen fabric adapts (throughout the arterial domain) to
restore its strain to the homeostatic value [see ( 17 ) and ( 21 )].
3.1 Case (i) Prescribed Elastin Degradation
First we illustrate a model of AAA evolution using a prescribed degradation of
elastin. As the aneurysm evolves (see Fig. 3 ) it can be seen that it develops a
preferential anterior bulging and becomes tortuous. Figure 3 a illustrates the
(prescribed) evolution of elastin concentration. In the central region of the AAA,
m E reduces to 0.05 whereas in the neck regions it is approximately 1. The average
collagen concentration (see Fig. 3 b) increases to compensate for loss of load borne
by the elastin and the increased load acting on the wall (due to the enlargement of
the geometry).
Figure 3 c and d depict the evolution of the elastin Green-Lagrange (GL) strains
E 11 and E 22 ; respectively. The strains are defined with respect to the unloaded
configuration, consequently they continue to increase with the enlargement of the
geometry: at t ¼ 0 ; E 11 ¼ 0 : 345 and E 22 ¼ : 281 whereas at t ¼ 10 ; maximum
values of E 11 ¼ 5 : 1 and E 22 ¼ 4 : 7 occur in the central region of the aneurysm.
Notice that E 11 decreases in the proximal section of parent artery as the aneurysm
enlarges; the axial expansion of the AAA is accentuated due to the axial retraction
of the ends of the artery.
The evolution of the collagen fibre GL strains differs substantially from that of
the elastin strains due to the evolution of the reference configurations that the
fibres are recruited to load bearing; recall this is achieved by remodelling the fibre
recruitment stretches [see ( 17 )] which define the factor the tissue must be stretched
in the direction of a fibre (relative to the unloaded reference configuration) for it to
begin to bear load. Figure 3 e depicts the evolution of the medial collagen GL
strain E M þ : At t ¼ 0 ; E M þ is constant throughout the domain with a magnitude of
the attachment strain, i.e. E M þ ¼ E AT ¼ 0 : 073 : As the AAA enlarges, the collagen
strains initially increase (at t ¼ 4 ; maximum values of 0.14 occur) and then reduce
as the aneurysm stabilises in size and the collagen fabric remodels to a material
equilibrium, i.e. E M þ ! E AT throughout the domain. Similar results are observed
for evolution of the adventitial collagen fibre GL strain E A þ (Fig. 3 f).