Biomedical Engineering Reference
In-Depth Information
• Case (i) Prescribed elastin degradation
The degradation of elastin is prescribed over a representative timescale of 10 years
(see [
16
]). To prescribe the loss of elastin, the following function is used:
"
#
2
exp
x
1
h
1
L
min
m
E
ð
h
1
;
h
2
;
t
Þ¼
1
1
m
min
ð
t
Þ
1
;
ð
13
Þ
where h
1
;
h
2
are the Lagrangian coordinates of the computational domain, i.e.
h
1
2
0
;
L
1
½
h
2
2
0
;
L
½ Þ;
m
min
represents the minimum concentration of elastin,
at time t
;
which, by construction, is located at
ð
h
1
;
h
2
Þ¼ð
L
min
;
h
2
Þ:
The parameter x
1
controls the degree of localisation of the degradation
function (we adopt x
1
¼
20
;
see [
16
] for influence of this parameter). We
assume that the minimum concentration decays exponentially. Thus,
m
min
ð
t
Þ¼
exp
f
ln
ð
m
T
Þð
t
=
T
Þg ¼ ð
m
T
Þ
t
=
T
;
ð
14
Þ
where T
¼
10
;
0
t
T
;
m
10
¼
0
:
05 and L
min
¼
0
:
8L
1
:
Note that at t
¼
0
;
m
min
ð
t
¼
0
Þ¼
1
:
• Case (ii) Linking Elastin degradation to Haemodynamic Environment
Initially, the distribution of the WSS is spatially uniform. To perturb the spatial
distribution of haemodynamic stimuli, a localised axisymmetric degradation of
elastin is prescribed to create a small axisymmetric AAA. We assume this small
aneurysm develops over several years and for numerical illustration we specifi-
cally consider a time-scale of 4 years. The functional form utilised for case (i) is
adopted, i.e, (
13
) with L
min
¼
0
:
5L
1
and m
4
¼
0
:
5
:
Whilst the elastin is degrading,
the aneurysm continues to enlarges in size and the strains in the collagen fabric are
elevated above homeostatic values, i.e. k
J
[ k
AT
). We stabilise the aneurysm by
switching off the degradation of elastin and allowing sufficient time for the col-
lagen fabric to adapt to a new homeostasis, i.e. k
J
p
!
k
AT
; numerically a period of
1 year or greater seems sufficient and so for 4\t
5
;
we specify that no further
elastin degrades (
8
t : 4\t
5
;
m
min
ð
t
Þ¼
m
4
). Subsequent degradation of elastin
(for t [ 5) is explicitly linked to deviations of haemodynamic stimuli from
homeostatic levels (see [
35
] for further details on methodology). The concentra-
tion of elastin m
E
evolves according to
o
m
E
ot
¼
F
D
D
max
m
E
ð
15
Þ
where t is in years, D
max
specifies the maximum rate of degradation, and
F
D
ð
h
1
;
h
2
;
t
Þ
: 0
F
D
1 is a spatially-dependent function of the haemo-
dynamic quantities to be linked to elastin degradation. Clearly, if F
D
¼
1
;
elastin degrades at a maximum rate whilst if F
D
¼
0
;
no degradation occurs.
As an illustrative example, we suppose elastin degradation is linked to low
levels of WSS. More specifically, we follow [
35
] and assume that if the
maximum value of the WSS is greater than a critical value, say s
crit
;
no
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