Biomedical Engineering Reference
In-Depth Information
or if the mixture velocity
v
is negligible (consistent with a quasi-static assumption
that is used for the structurally significant constituents)
∂ρ
i
∂τ
=¯
div
j
i
.
m
i
−
(9.4)
Noting that we have
N
n
constituent mass densities
ρ
i
, we clearly must introduce additional (constitutive) relations for
−
n
equations to determine
N
−
m
i
and
j
i
.For
dilute solutions, the mass flux for diverse molecular species is often approximated
by Fick's law, which is typically written in terms of molar, not mass, densities. Note,
therefore, that the molar density
C
i
¯
ρ
i
/MW
i
where
MW
i
are molecular masses.
Hence, the mass balance equation for the soluble constituents can be written
≡
∂C
i
∂τ
=
div
J
i
,
R
i
−
(9.5)
where
C
i
are also called concentrations,
R
i
are reactions responsible for produc-
tion/removal, and by Fick's law
J
i
D
i
grad
C
i
, where
D
i
are the diffusivities.
Hence, we obtain the classical reaction-diffusion equation
=−
∂C
i
∂τ
=
R
i
D
i
2
C
i
,i
+
∇
=
1
,
2
,...,N
−
n
(9.6)
for all soluble constituents at G&R times
τ
.
The situation is very different for the insoluble, structurally significant, con-
stituents, i.e. Eq. (
9.2
). We previously introduced an additional assumption that all
structural constituents are constrained to move with the mixture (Humphrey and
Rajagopal,
2002
). This assumption coupled with the quasi-static assumption thus
requires that the motions
x
α
∈[
0
,s
]
=
x
=
0
, whereby velocities are similarly constrained:
v
α
=
v
=
0
. Equation (
9.2
) thus can be written
∂ρ
α
∂τ
=¯
∂ρ
α
∂τ
d
τ
m
α
m
α
d
τ.
or
=
¯
(9.7)
We thus have
n
equations to determine
n
mass densities, which again necessitates
the introduction of additional (constitutive) relations for the net production/removal
function. Yet, because
m
α
0 during periods of tissue maintenance (i.e., balanced
production and removal in unchanging configurations), we have shown previously
that it is convenient to assume a separable representation
¯
=
m
α
(τ)q
α
(s,τ)
,
where
m
α
(τ)>
0 is the true rate of mass density production and
q
α
(s,τ)
m
α
(τ)
¯
=
]
is a survival function that accounts for the fact that all cells and proteins have a
finite half-life (Valentín et al.,
2009
). Hence, the survival function represents the
percentage of constituents produced at time
τ
that survives to current time
s
.
It can be shown that use of the separable form for the net production term allows
Eq. (
9.7
) to be written in a reduced form, namely
∈[
0
,
1
s
ρ
α
(s)
ρ
α
(
0
)Q
α
(s)
m
α
(τ)q
α
(s,τ)
d
τ,
=
+
∀
α
=
1
,
2
,...,n,
(9.8)
0
