Biomedical Engineering Reference
In-Depth Information
o
N
chn
ot
¼
1
r
o
or
rvN
ch
ð Þ
b
chn
N
chn
ð
N
chn
N
chn
Þþ
k
cm
f
ð
r
Þ
C
tgf
N
msc
d
chn
C
tnf
N
chn
;
ð
15
Þ
for R
c
r
R
o
:
Chondrocytes are assumed not to be able to migrate, hence there
is no diffusion term in Eq. (
15
) and no boundary conditions need be specified.
IL-10 is secreted inside the trachea by MSCs, where it diffuses and undergoes
degradation:
þ
b
ilm
N
msc
d
ilk
C
ilk
;
o
C
ilk
ot
¼
1
r
o
or
rvC
ilk
þ
rD
ilk
o
C
ilk
or
ð
16
Þ
for R
l
r
R
o
:
The boundary conditions stipulate that IL-10 is secreted into the
trachea by EPCs on the lumenal surface at a rate proportional to EPC density, but
does not diffuse out of the trachea at r
¼
R
o
:
D
ilk
o
C
ilk
or
oC
ilk
ot
ð
R
l
;
t
Þ¼
b
ilk
N
epi
;
ð
R
o
;
t
Þ¼
0
:
ð
17
Þ
TGF-b1 is produced by macrophages and MSC-derived chondrocytes and dif-
fuses through the domain while being degraded
o
C
tgf
ot
¼
1
r
o
or
rvC
tgf
þ
rD
tgf
oC
tgf
or
ð
18
Þ
þ
b
tch
N
chn
ð
N
chn
N
chn
Þþ
b
tm
N
mac
d
tgf
C
tgf
;
for R
l
r
R
o
:
It is assumed that TGF-b1 is produced by the chondrocytes only
while they are proliferating i.e. until their numbers reach the carrying capacity,
hence the rate of production of TGF-b1 by these cells is assumed to be propor-
tional to the difference between the current density and the carrying capacity, N
chn
:
Zero-flux boundary conditions are applied to the concentration of TGF-b1 at the
lumenal and outer surfaces:
o
C
tgf
or
ð
R
l
;
t
Þ¼
o
C
tgf
ð
R
o
;
t
Þ
or
¼
0
:
ð
19
Þ
TNF-a is released by macrophages at a rate that is decreased by IL-10, diffuses
through the trachea and is degraded. Hence the equation governing the concen-
tration of TNF-a is
þ
b
tnf
K
ti
K
ti
þ
C
ilk
o
C
tnf
ot
¼
1
r
o
or
rvC
tnf
þ
rD
tnf
o
C
tnf
or
N
mac
d
tnf
C
tnf
;
ð
20
Þ
for R
l
r
R
o
and with zero-flux boundary conditions at the edges of the domain:
o
C
tnf
or
ð
R
l
;
t
Þ¼
o
C
tnf
or
ð
R
o
;
t
Þ¼
0
:
ð
21
Þ
Search WWH ::
Custom Search