Biomedical Engineering Reference
In-Depth Information
11.3.2.3
Boundary and Initial Conditions
In addition to the boundary conditions provided in Section 11.2.2.4, at the
center of the cartilage (i.e., at
r
= 0),
∂c
I
∂r
= 0
(11.44)
r
=0
Likewise the following additional initial condition is assumed:
c
I
(
r,
0) = 0
(11.45)
That is, we have assumed that initially there is no bound growth factor
within the tissue.
11.3.3 Results and Discussion
Since the solute concentration is generally n
on
uniform along the radial direc-
tion, it is useful to define an average total
(
R
U
) IGF-I uptake ratio, which is
the su
m
of the average free uptake ratio (
R
f
) and the average bound uptake
ratio (
R
b
),
R
u
=
R
f
+
R
b
(11.46)
where
R
f
=
r
0
2
πrc
I
dr
R
b
=
r
0
2
πrc
I
dr
and
c
I
0
r
0
(11.47)
c
I
0
r
0
2
πrdr
2
πrdr
0
0
Here
c
I
0
is the solute concentration at the outer edge of cartilage disc, and
c
I
0
is the initial volume-based bound solute concentration. Simply stated,
the uptake ratios describe the average IGF-I concentration within the car-
tilage (free, bound, or total) relative to the free IGF-I concentration in the
surrounding IGF-I bath solution.
11.3.3.1
Free Diffusion
Figure 11.6 allows comparison between the numerical predictions of free dif-
fusion (with and without consideration of binding) and the experimental data
of Bhakta et al. [29]. It can be seen that the model is able to produce a rea-
sonable fit to the steady-state experimental data when the model includes
binding of IGF-I to IGFBPs with an IGFBP concentration
c
BP
0
= 45 nM.
If the bath IGF-I concentration is low, ignoring binding results in signifi-
cant underestimation of total IGF-I uptake; however, the influence of binding
becomes negligible if the concentration is high due to the saturation of the
finite number of binding sites.
Figure 11.7 shows the total solute uptake ratio profiles at time intervals
of 5 h, with and without binding (40 nM bath concentration). It can be seen
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