Biomedical Engineering Reference
In-Depth Information
the second mechanical damage variable
d
2
is defined as a function of accumulated
equivalent strain through the accumulation variable
α
2
(
t
)
,
t
dW
o
ds
α
2
(
t
)=
ds
.
(6.55)
0
The corresponding evolution equation is,
=
)
W
o
α
02
˙
(
t
(6.56)
with the initial condition
α
2
(
0
)=
0. For this damage mode,
α
2
accumulates contin-
uously as the material deforms.
Enzymatic damage
We also consider the possibility that damage is accumulated due to enzymatic (or
other chemical) damage. For example, in arteries, abnormally high wall shear stress
under suitable wall shear stress gradients has been associated with damage to the
internal elastic laminae of arteries [33, 46, 64, 81]. Animal studies have shown that
this degradation can be induced by exposing bifurcations to abnormal hemodynamic
loads. In particular, the combination of elevated wall shear stress
WSS
and positive
elevated wall shear stress gradient
WSSG
has been associated with elastin degrada-
tion in native and nonnative bifurcations [76, 77]. The degradation has been found
to be progressive in that damage increases with exposure time [81]. The degree of
pre-aneurysmal type change was found to be dose dependent. Motivated by these
results, the following form for
α
was proposed [74],
α
3
(
t
)=
f
(
WSS
,
WSSG
)
(6.57)
where the appropriate definition of
WSSG
is discussed in [120]. This form of
α
3
is
purposely left quite general, but will be similarly chosen to be an invariant quantity.
A particular example is
t
0
H(
ζ
)H(
η
)(
ζ
+
1
T
α
3
=
b
η
)
ds
where
WSS
WSS
T
WSS
T
−
WSSG
WSSG
T
WSSG
T
−
ζ
=
,
η
=
.
(6.58)
The corresponding evolution equation is
1
T
H(
ζ
(
˙
α
3
=
t
))H(
η
(
t
)) (
ζ
(
t
)+
b
η
(
t
)))
(6.59)
where
b
and
T
are material constants. Eq. (6.59) satisfies the criterion that damage
does not increase unless WSS is sufficiently elevated above a threshold value and the
WSSG is elevated above a positive threshold value. Further, the evolution Eq. (6.59)
satisfies the condition that damage increases with exposure time and dosage. More
data is needed to determine whether the rate of accumulation should increase with
increased
η
(amount by which WSSG exceeds the threshold value) or whether it
