Biomedical Engineering Reference
In-Depth Information
4 μQ/πa 3 each near target/homeostatic values (e.g., σ θ and τ w , respectively,
where P,a,h,μ and Q are blood pressure, luminal radius, wall thickness, blood
viscosity, and volumetric flow, respectively). As shown previously (Humphrey,
2008a ), if we let parameterize the change in blood pressure from normal and pa-
rameterize the change in blood flow from normal (e.g., γ =
τ w =
1 . 5 for a 50 % sustained
increase in pressure), then it is easy to show that a ε 1 / 3 a h and h γε 1 / 3 h h
(where the subscripts h denote homeostatic values) to maintain/restore the stresses
to homeostatic targets in response to modest alterations in blood pressure or flow.
Whereas these simple relations describe the extent of the morphological adaptations,
they cannot describe the time-course of such changes or the associated changes in
structure or properties. In contrast, the G&R framework described by Eqs. ( 9.8 ) and
( 9.12 ) can address both the extent and rate of each of these changes.
Fundamental to geometric and structural changes in arteries are changes in rates
of turnover of structurally significant constituents such as the smooth muscle and
fibrillar collagens. For example, we have shown that the following constitutive rela-
tions (cf. Eq. ( 9.14 )) provide a good description of large artery adaptations to both
altered blood pressure and flow:
m B 1
K τ w τ w ,
m α (τ)
K σ σ
=
+
(9.15)
exp
τ ,
s
K q 1
+ σ(τ) 2 d
q α (s τ) =
(9.16)
τ
where the stress differences are given by
σ h
σ h
τ w
σ
τ w
σ
=
,
τ w =
,
(9.17)
τ w
with σ an appropriate scalar metric of intramural stress. Note, too, that the gain-
type parameters K in Eqs. ( 9.15 ) and ( 9.16 ) modulate the stress-mediated changes in
mass production and removal. Although these particular functional forms are among
the simplest possible, basal rates are recovered ( m B and K q ) when the stresses equal
their homeostatic targets, as desired, and associated simulations have captured many
salient aspects of observed adaptations (Valentín and Humphrey, 2009a , b ). Note,
too, that the survival function recovers first order kinetic decays as suggested by
much of the data (cf. Humphrey, 2008b ).
At this juncture, it is important to recognize that these constitutive relations are
motivated by mechanobiological observations, yet they are phenomenological. For
example, it is well known that collagen synthesis is increased by increases in cyclic
stretch/stress of smooth muscle cells. It is also well known that increases in wall
shear stress increase endothelial cell production of the vasodilator nitric oxide (NO)
and decreases in wall shear stress increase endothelial cell production of the vaso-
constrictor endothelin-1 (ET-1); see Fig. 9.1 and Humphrey ( 2008b ). Moreover, NO
decreases the production of collagen by smooth muscle cells whereas ET-1 increases
the production rate (hence the minus sign in Eq. ( 9.15 ) for the shear stress mediated
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