Biomedical Engineering Reference
In-Depth Information
right Cauchy-Green tensor:
C
n(τ)
(s)
. In particular,
n(τ)
reminds us that this defor-
mation is referred to the natural configuration
κ
n
(τ)
for that individual constituent
at its time of deposition
τ
. To appreciate the assumed form in Eq. (
9.12
),
note that if there is no G&R, then
s
∈[
0
,s
]
=
0 and this equation reduces to
ρ
α
(
0
)Q
α
(
0
)
ρ(
0
)
W
α
C
n(
0
)
(
0
)
=
φ
α
(
0
) W
α
C
n(
0
)
(
0
)
W
α
(
0
)
=
(9.13)
(recalling that
Q
α
(
0
)
1 by definition), which recovers a simple rule-of-mixtures
relation as desired. It can be shown similarly that the simple rule-of-mixtures rela-
tion is recovered in the case of tissue maintenance, that is, balanced production and
removal in unchanging configurations (Valentín et al.,
2009
).
Most importantly, Eqs. (
9.8
) and (
9.12
) reveal the need to determine three ba-
sic types of constitutive relations for each structurally significant constituent
α
≡
=
1
,
2
,...,n
, namely
W
α
C
n(τ)
(s)
.
m
α
(τ),
q
α
(s
−
τ),
(9.14)
In our prior implementations (e.g., Baek et al.,
2006
; Valentín et al.,
2009
), we have
used phenomenological constitutive relations motivated by tissue level observations
of mechanobiological responses by arteries in response to diverse mechanical loads
(Humphrey,
2008b
). For example, we have modeled the energy stored in the elastin
dominated amorphous matrix using a classical neo-Hookean relation and the en-
ergy stored in collagen fibers and passive smooth muscle using classical Fung-type
exponential relations. For the present discussion, it is important to note that the
neo-Hookean relation was first derived based on micromechanical arguments and
exponential relations have been shown to capture well the net mechanical response
of collections of fibers having linear behaviors but a distribution of undulations. It is
suggested that increased attention should be given to the derivation of microstruc-
turally based constitutive relations for the energy stored in individual constituents
as well as interaction energies between constituents. Such relations would enable
better modeling of many disease processes wherein either particular constituents
are absence because of genetic mutations (e.g., fibrillin-1, which stabilizes elastic
fibers, or collagen III, as in Marfan and Ehlers-Danlos IV syndromes, respectively)
or chemomechanical injury (e.g., degradation or fatigue of elastic fibers in aging).
Below, however, let us focus on constitutive relations for mass production and re-
moval, which are unique to G&R theories.
9.3 Towards Multiscale Constitutive Relations
Two of the best studied arterial adaptations are responses to sustained alterations
in blood pressure and flow, the former of which is particularly relevant to hyper-
tension research. It is well accepted that large arteries tend to grow and remodel
so as to keep the mean circumferential stress
σ
θ
=
Pa/h
and the wall shear stress
