Multi-Scale Modelling of Vascular Disease: Abdominal Aortic Aneurysm Evolution - Computational Modeling in Tissue Engineering - page 313

up-regulation of ECM protein expression and a down-regulation of collagenase
expression. Conversely, relaxation of the substrate can trigger different signals
resulting in a reversed pattern of protein expression [ 59 ], i.e. down regulation of
ECM protein expression and up-regulation of collagenase expression. Although
we do not explicitly model protein synthesis and enzymatic degradation, the net
result is to stimulate increases/decreases in the ECM. The simplest functional form
for F G that satisfies these requirements is linear,and hence we propose the fol-
lowing evolution equation for the collagen fibre concentration

!

dm J p
dt

E J p j max E AT

E AT

¼ b ð A CS Þ m J p

; ð 21 Þ

where b ð A CS Þ is a phenomenological growth parameter. For a healthy abdominal
aorta, vascular cells experience cyclic stretching with magnitudes of approxi-
mately 10 % whilst for older, more collagenous vessels, the magnitudes may
decrease to 2 % [ 62 ]. We select a functional form for b which increases rates of
production if the cyclic stretch environment increases from its initial values for the
healthy artery at t ¼ 0:

:

A CS
A CS j t ¼ 0

b ð A CS Þ¼ b 0 exp b CS max 1 ; 0 ð 22 Þ

This choice is somewhat arbitrary, however it acts to prevent unrealistically large
cyclic deformations that can occur in regions of the tissue as the aneurysm evolves.
For the analysis in this paper, we follow [ 37 ] and take b 0 ¼ 0 : 7 years 1 ; b CS ¼ 3 :
Average fibre concentration
To simplify the presentation of the results, we define the average fibre con-
centration m C of the medial and adventitial layers,where

!

m M þ þ m M
2

m A þ þ m A
2

m C ¼ 1
H

þ H A : ð 23 Þ

H M

3 Physiological Growth Model Abdominal Aortic Aneurysm

We briefly review the framework of the model prior to presenting the results.
Equation ( 2 ) defines the (spatially and temporally heterogeneous) SEFs for the
medial and adventitial layers of the aneurysmal tissue. An AAA develops as the
material constituents of the artery evolve. The variation of the total potential
energy ( 1 ) governs the equilibrium displacement field, and is solved by the finite
element method [ 15 ]; volume meshes of the luminal computational domain are
generated and the haemodynamics are solved using finite volumes [ 35 ]. In many
aneurysms, diameter enlargement is asymmetric, with primarily anterior