Biomedical Engineering Reference
In-Depth Information
variable; the concentration of RANK is kept fixed to reflect the undiminished
availability of the osteoclast precursors) are the following [2,3]:
dO
dt
=−⋅
pkOL kOLd
O
⋅ +⋅ •−
(6.5)
1
2
O
(
)
dO L
dt
•
=⋅ ⋅− ⋅ •
kOLkOL
1
(6.6)
2
dL
dt
(6.7)
=−⋅
pkOL kOLkKL kKLd
L
⋅ +⋅ •− ⋅
⋅ +⋅ •−
1
2
3
4
L
(
)
dK L
dt
•
=⋅ ⋅− ⋅ •
(6.8)
kKLkKL
3
4
where the meaning and values of the parameters
k
1
~
k
4
can be found in
Table 6.1 on the previous page, and
d
o
=
k
o
·
O
is a linear degradation function,
with
k
O
being the rate of elimination of OPG.
In the analysis, Wang et al. [2] and Wang [3] proposed that cell prolifera-
tion rate is proportional to the receptor occupancy [25] and they applied this
rule to other types of cell response in addition to cell proliferation. Moreover,
the antiproliferative cell responses are inversely proportional to the receptor
occupancy. Consequently, the production rate of OPG (
p
o
) is downregulated
by PTH and upregulated by TGF-β, and the expression of
p
o
is
1
OO
P
pK
=⋅
+π
RI
+
O
(6.9)
C
π
P
where
K
P
is the minimal production rate of OPG per cell,
I
O
is a rate of exter-
nal injection of OPG, and at equilibrium state the proportion of occupied
PTH receptor π
P
is
π=
•
PP
R
P
PP
r
=
(6.10)
P
P
s
+
T
s
0
PkkP SI
=
/,
=
(
+
)/
kPP
=
+
,
Considering
with
P
0
=
S
P
/
k
P
and
6
5
PP P
P
= S
P
/
k
P
,
at equilibrium state, we have
0
π≈
+
+
PP
PP
Ik Sk
Ik kk
+
+
PP
PP
=
(6.11)
P
s
PP
65
for
P
0
<<
P
S
.
As mentioned in Lemaire et al. [21], unlike OPG, RANKL is attached to
osteoblast surfaces and occupies a restricted space. The surface of osteoblasts
