Biomedical Engineering Reference
In-Depth Information
models that can predict time-dependent changes in composition, structure, geom-
etry, and properties that occur in response to changes in the biochemomechanical
environment. Although much more data will be needed to model precisely many
of the underlying mechanisms that are responsible for such growth and remodeling
(G&R), expanding data bases provide sufficient guidance on salient aspects of de-
velopment, adaptation, and disease progression for us to begin to interpret these data
within mathematical frameworks. Toward this end, here we consider a constrained
mixture model of tissue-level arterial adaptations that can incorporate molecular in-
formation related to the underlying mechanisms. Areas requiring further research
are then highlighted to encourage continued development of these models.
9.2 Continuum Framework
By growth, we mean a change in mass; by remodeling, we mean a change in struc-
ture. Notwithstanding the many associated complexities at different spatial and tem-
poral scales, we begin by assuming that G&R occurs via quasi-static isothermal
processes, which focuses our attention on equations of mass balance and linear mo-
mentum balance. Moreover, let us assume that the arterial wall can be modeled
as a mixture consisting of
N
constituents, including
α
=
1
,
2
,...,n
insoluble but
structurally significant constituents and
i
n
soluble but structurally
insignificant constituents. Examples of the former are elastic fibers, fibrillar colla-
gens, muscle fibers, and proteoglycans; examples of the latter include vasoactive,
mitogenic, proteolytic, and inflammatory molecules. We have previously discussed
the utility of employing full mixture equations to describe mass balance for both
classes of constituents, but a rule-of-mixtures relation for the stress response that
can be used to satisfy overall linear momentum balance (Humphrey and Rajagopal,
2002
).
Mass balance, in spatial form, can be written as
=
1
,
2
,...,N
−
∂ρ
i
∂τ
+
div
ρ
i
v
i
=¯
m
i
,i
=
1
,
2
,...,N
−
n,
(9.1)
∂ρ
α
∂τ
+
div
ρ
α
v
α
=¯
m
α
,α
=
1
,
2
,...,n,
(9.2)
where
ρ
i
and
ρ
α
are so-called apparent mass densities (constituent mass per mix-
ture volume) and
m
α
are the so-called net rates of mass density produc-
tion/removal (which can be positive, zero, or negative);
τ
m
i
and
∈[
0
,s
]
is the G&R time,
which is typically much greater than the cardiac cycle timescale
t
.
Focusing first on the
N
−
n
soluble constituents, i.e. Eq. (
9.1
), it is convenient
to introduce the mass flux
j
i
ρ
i
(
v
i
v
)
where
v
i
v
is sometimes called the
'diffusion velocity.' Regardless, Eq. (
9.1
) can be written at G&R time as
=
−
−
∂ρ
i
∂τ
+
div
ρ
i
v
=¯
div
j
i
,
m
i
−
(9.3)
