Biomedical Engineering Reference
In-Depth Information
where
D
A
is a constant degradation rate of substrate
A.
Pivonka et al. further
assumed that endogenous production is regulated by a ligand and that pro-
duction cannot exceed a maximum level of concentration (
A
max
):
A
A
Pt
()=⋅
βΠ 1
⋅ −
( 7. 2 3)
Ae
,
Aact rep
/
A
max
Substituting Equations (7.22) and (7.23) into Equation (7.20) gives
A
=
β⋅Π+
β⋅Π
Pt
()
Aact
/
rep
Ad
,
A
( 7. 2 4)
A
Aact
/
rep
+
D
A
A
max
which can be used to derive the expression of PTH. Pivonka et al. [3] took
PTH as a regulator of RANKL and OPG production. The assumptions were
made that PTH endogenous production was constant (i.e., βPTH = constant
and
Π= 1)
act
PTH
/
max
,
and that the binding of PTH to its recep-
tors on OBP and OBA was the same (
N
= 1), to obtain PTH concentration and
its according activation and repression functions. Applying these assump-
tions to Equation (7.24), PTH can be obtained as [3]
and
PTH
>>
PTH
rep
(
)
=
β+
Pt
PTH
PTH d
,
PTH
( 7. 2 5 )
D
PTH
where β
PTH
is the synthesis rate of systemic PTH,
P
PTH, d
(
t
)
represents an exter-
nal PTH dosing term, and
D
PTH
is the rate of degradation of PTH. Making
use of Equation (7.25), the activator/repressor input functions defined in
Equations (7.3) and (7.4) can be calculated as
PTH
PTH
PTH
PTH
Π=
and
Π=
( 7. 2 6 )
actOBP
,
actOBA
,
K
+
PTH
K
+
PTH
D
4,
PTH
D
5,
PTH
1
1
PTH
PTH
and
Π=
Π=
( 7. 2 7 )
rep OBA
,
rep OBP
,
1
+
PTH K
1
+
PTH K
D
6,
PTH
D
7,
PTH
where
K
D
4,
PTH
/
K
D
5,
PTH
are activation coefficients for RANKL
eff
production on
OBP/OBA related to PTH binding.
K
D
6,
PTH
/
K
D
7,
PTH
are the repression coeffi-
cients for OPG production related to PTH binding on OBP/OBA. As a first
approximation, Pivonka et al. assumed that the activation and repression
coefficients for different osteoblastic cells were the same (i.e., where
K
D
4,
PTH
=
K
D
5,
PTH
and
K
D
6,
PTH
=
K
D
7,
PTH
) and hence had the same activator/repressor
function.
