Biomedical Engineering Reference
In-Depth Information
In other words, rather than purely phenomenological forms represented by
Eq. (
9.15
), molecular level information could be used to inform the continuum level
analysis. Similar relations could be determined for the survival function, which
should include terms accounting for concentrations of active proteases. Moreover,
relations for changes in molecular production can be derived from appropriate ex-
periments, as, for example, studies of the effects of changing wall shear stress on
the production of NO, as, for example (Humphrey,
2008b
)
C
N
B
ζ
β
1
exp
−
δτ
w
,
C
NO
=
+
−
(9.20)
where (
ζ,β,δ
) are parameters required to fit the data (e.g.,
ζ,β,δ
=
0
.
37
,
0
.
63, and
−
8
.
89 as reported in Humphrey,
2008b
). In this way, molecular level (mechanistic)
relations can be combined simply with continuum level models that have already
proven useful in modeling diverse aspects of arterial G&R.
9.4 Discussion
Bioengineers and clinicians must similarly address arterial adaptations at a macro-
scopic scale—including normal changes due to development or exercise as well as
disease progression, responses to treatment, and so forth. Classical examples include
quantification of wall thickening and stiffening in hypertension, changing caliber
in exercise or arterio-venous fistulas, stenoses in vein grafts, evolving atheroscle-
rotic plaques, aneurysms, and so forth (Taylor and Humphrey,
2009
). Continuum
level biomechanical modeling has proven fundamental to studying such tissue-level
changes and will likely remain so for purposes of diagnosis, interventional plan-
ning, medical device design, and many other daily activities. Nevertheless, we must
also exploit our growing understanding of the molecular level mechanisms that dic-
tate macroscopic manifestations. We submit here that a consistent mixture theory
framework for growth and remodeling allows one to account naturally for spatial
and temporal changes in effector molecules via classical reaction-diffusion equa-
tions, which in turn can be used to inform improved constitutive relations for cell
and matrix turnover that are fundamental to the tissue-level analyses that are vi-
tal for so many aspects of research and clinical care. Indeed, we emphasize that the
present G&R framework, which focuses on changes to the arterial wall, is also easily
coupled to sophisticated computational fluid dynamics simulations of the hemody-
namics (Figueroa et al.,
2009
), thereby permitting both multiscale and multi-physics
studies. Moreover, we emphasize that the multiscale approach presented here (fo-
cused mainly on informing continuum level constitutive relations with molecular
level information) is but one possible multiscale approach. Hayenga et al. (
2011
),
recently showed that agent based models can similarly be integrated with continuum
level G&R models, hence providing yet another level of multiscale modeling.
In summary, there is a pressing need for continued research on the molecular
mechanisms responsible for arterial adaptations and disease progression, particu-
larly given the complex multifunctional capabilities of the large number of effector
