Biomedical Engineering Reference
In-Depth Information
can carry only a limited number of RANKL molecules. Therefore, the
RANKL production rate
p
L
should be self-limiting to prevent the number of
RANKL from exceeding the carrying capacity of the osteoblast surface. The
effective carrying capacity is imposed by the cell response to PTH binding,
although there is a possible maximum number of RANKL attached on each
cell surface, noted as
K
.
L
P
Applying the same rule as that for obtaining
P
O
,
the
control of RANKL concentration on the cell surface is obtained as [21]
−=⋅ −
+•+•
⋅π ⋅
LOLKL
K
+
pdr
1
I
(6.12)
L
L
L
L
P
B
L
P
where
r
L
is the rate of RANKL production and elimination, and
I
L
is a rate
of external injection of RANKL.
L
+
O
•
L
+
K
•
L
is the current total con-
centration of RANKL, where the denominator corresponds to the imposed
carrying capacity. In Equation (6.12), the concentration of RANKL that the
osteoblasts aim to produce is governed by the value of π
P
:
P
P
.
If the
total concentration of RANKL is below this value, the rate of production
of RANKL is positive; otherwise, it is negative. Pseudosteady states of the
RANKL and OPG concentrations are calculated from Equations (6.5)-(6.9)
and (6.12):
⋅π
KB
L
P
KB
kKkkOk
⋅π ⋅
I
r
L
P
L
L
(6.13)
L
=
⋅ +
1
1
+
+
3
4
1
2
P
K
I
k
O
O
O
=
R
+
(6.14)
k
π
OP
O
The osteoblast lineage is supplied from a large population of uncommitted
progenitors. These progenitors express a specific TGF-β receptor that, once
activated, leads to the differentiation of the progenitors into ROBs. If the rate
of release of TGF-β per active osteoclast is constant, the entering flow into the
ROB compartment depends only on the mesenchymal stem cells' response
to c binding. This response is represented by a proportionality relationship
with the TGF-β receptor occupancy π
C
:
0
CC
CC
+
+
D
⋅π =⋅
D
(6.15)
RC
R
s
where
D
R
is a proportionality factor
C
is the concentration of the AOCs
C
s
corresponds to the dissociation binding coefficient of TGF-β with its
receptor
