Biomedical Engineering Reference
In-Depth Information
hydration reduces, the volume containing the fixed charges decreases due to a geo-
metrical effect. This model provides an accurate swelling pressure prediction over
all ranges of hydration. The model is further improved by partitioning the charge
into two GAG-based groups: one associated with a charge-dense coating of the
collagen fibrils, in accordance with experimental evidence, and one dispersed over
the interfibrillar volume and associated with GAGs which bridge fibrils, possibly
by duplexing. This refinement gives a surprisingly accurate prediction for swelling
pressure at
λ
0
.
6, when 40 % of the charge is in the coating region. This result
implies that around physiological hydration (
t
=
=
0
.
5 mm), the swelling pressure pri-
marily results from the bridging GAGs in the interfibrillar region. As the tissue is
compressed toward
t
∼
0
.
25 nm, the coatings start to play a role by mutually inter-
acting and producing a local increase in charge density and a concomitant increase
in swelling pressure.
The third model employs a molecular-level unit cell in which volumetric do-
mains within the unit cell are associated with the macromolecular GAGs. The model
is a hybrid approach in that it represents the fibril coating as a charge-dense re-
gion around the fibrils using a continuum description. The approach accounts for
the spatially varying electrostatic potential between the explicit GAG domains and
the mobile ions, again using the Poisson-Boltzmann equation. The results of this
model applied to the case of next-nearest neighbor GAG connectivity, as proposed
by Muller et al. (
2004
), are also quite accurate. This approach does introduce ad-
ditional variables that are not readily estimated, including the GAG length ratio
α
,
which describes the waviness of the GAG chains, and the radius of the effective
GAG cylinder
r
g
. However, the length ratio
α
for the bridging GAGs is also a rel-
evant parameter for estimating the entropic elasticity of the polymer chain using a
theory such as the wormlike chain model.
In the present study, we have addressed the swelling problem in terms of the
electrostatic component of the free energy alone. However, the chemomechanical
free energy, which includes the entropic elasticity of the GAGs and the molecular
mixing energy, will certainly have some influence on the swelling pressure (Hart
and Farrell,
1971
; Jin and Grodzinsky,
2001
). Indeed, the GAG entropic elasticity
will produce expansion-resisting forces that will contribute a 'negative' swelling
pressure component (Hart and Farrell,
1971
). These non-electrostatic components
will be the subject of a future study.
Acknowledgements
This research was supported by the Stanford University Bio-X Interdisci-
plinary Initiatives Program (PMP), which is gratefully acknowledged. XC also received support
from a Stanford Graduate Fellowship.
References
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