Biomedical Engineering Reference
In-Depth Information
described previously, the synovial membrane is the tissue that performs the filtering func-
tion. This membrane is composed of two layers, the synovium, which is approximately
50
μ
μ
m thick. The synovium is
the layer that is composed of an incomplete layer of macrophages and fibroblasts, which
prevents foreign particles from entering the joint space and it produces some of the mole-
cules that comprise synovial fluid. The sub-synovium is composed of extracellular fibers
that filter the interstitial fluid. The primary extracellular fibers within this layer are various
collagens, fibronectin, hyaluronan, and various proteoglycans. What is interesting about
the sub-synovium is that it is permeable enough to allow fluid to enter the joint space but
exhibits a high enough resistance to prevent fluid from exiting the joint space (i.e., it acts
as a uni-directional valve). Once the synovial fluid enters the joint space, it is associated
with the charged proteins within the cartilage and only “flows” under a compressive load
of the joint. The modeling of this flow is quite complex and will be described in the
remainder of this section.
The synovial membrane must be modeled as a semipermeable membrane that has a dif-
ferent permeability toward different molecules within the synovial fluid. Also, cartilage
would need to be modeled as a store for the lubricant and molecules, which may become
available during loading conditions. The time rate of change of molecule
i
within the syno-
vial fluid would be represented as
m thick and the sub-synovium, which is approximately 100
@ð
Vc
Þ
i
@t
5
ðr
s
A
s
1
r
c
A
c
2
d
2
JÞ
i
ð
11
:
1
Þ
where
V
is the volume of the synovial fluid,
c
is the concentration of molecule
i
,
r
is the
formation rate of
i
by the synovial membrane (denoted with a subscript s) or the cartilage
(denoted with a subscript
c
),
d
is the degradation rate of molecule
i,
and
J
is the flux of
i
out of the synovial membrane. The degradation rate is typically dependent on the concen-
tration and kinetic rate constants of degradation enzymes within the synovial fluid. The
flux of molecules out of the synovial membrane is dependent on the permeability of the
membrane, the concentration gradient, the cross-sectional area of the membrane, and the
ratio of restricted diffusion to free diffusion. For this type of diffusion, we would need to
develop a restricted diffusion coefficient, because of the interaction of the molecules with
the charged proteins within synovial fluid and the cartilage. The restricted diffusion coeffi-
cient can be defined as
p
Þ
ð
a
a
Þ
D
i
5
De
ð
2
1
1
ð
11
:
2
Þ
where
D
is the free diffusion coefficient of species
i, θ
is the volume fraction of the protein
molecules (e.g., proteoglycans),
a
i
is the effective radius of the molecule
i
, and
a
is the
effective radius of the protein. The effective radii can be calculated from the Stokes-
Einstein formulation (which is valid for any molecule):
1
=
3
3MW
4
a
x
5
ð
11
:
3
Þ
πρN
A
where MW is the molecular weight,
is the density, and
N
A
is Avogadro's number. From
these formulas, it is possible to model the changes in the concentration of various species
ρ
Search WWH ::
Custom Search