Biomedical Engineering Reference
In-Depth Information
of
z
through binding to their respective receptors on the same cell. Then the
production rate of
z
can be expressed as
(
)
∗
∗
∗∗
x
y
zf xy
=
(, )
=β⋅Π+
k
k
⋅Π
( 7. 5 )
x
act
/
rep
y
act
/
rep
where
K
x
and
K
y
are the relative influence of ligands
x
and
y,
respectively, as
a percentage in the cellular process, and
K
x
+
K
y
= 1.
For convenience, in the following related formulation, the abbreviated
forms are used for the factors involved. As in Figure 7.1, Qin and Wang [2]
used OBU for uncommitted osteoblastic progenitors, OBP for preosteoblast,
OBA for mature osteoblast, OCP for osteoclast precursor, OST for osteocyte,
and OCA for active osteoclasts. RL is used for RANKL, RK for RANK, Tβ for
TGF-β, and
P
2 for PGE
2
; OPG, NO, and PTH remain unchanged.
The equations governing the evolution of the number of osteoblastic and
osteoclastic cells in each maturation stage are simply balance equations [23],
which means that each cell stage is fed by an entering flow and is emptied by
the outgoing flow of differentiated or apoptotic cells (Figure 7.1). As a result,
utilizing Figures 7.1 and 7.2 and based on the formulation in Pivonka et al.
[3], the bone cell population dynamics can be written as follows:
dOBP
dt
(
)
T
β
P
2
T
β
=
Dk
⋅
⋅Π +⋅Π−
k
D
⋅
OBP
⋅Π
( 7. 6 )
OBUT
actOBU
β
P
2
act OBU
,
OBP
,
rep OBP
,
dOBA
dt
T
β
=
D
⋅
BP
⋅Π
−
A
⋅
BA
( 7. 7 )
OBP
OBA
rep OBP
,
dOST
dt
=
T
⋅
BA
−
A
⋅
ST
( 7. 8)
OBA
OST
dOCA
dt
RL
⋅Π
β
T
=
D
⋅Π−
A
⋅
OCA
( 7. 9)
OCPact OCP
,
OCA
actOCA
,
where
k
T
β
,
k
P
2
are the relative influence constants of TGF-β/PGE
2
bindings in OBU
differentiation
Π
β
actOBU
T
2
are the activator functions related to, respectively,
TGF-β and
P
2 bindings to their receptors on OBU
P
and
Π
actOBU
,
,
Π
β
rep
T
and
Π
β
actOCA
T
,
are the repressor/activator functions related to TGF-β
binding to its receptors on OBP and OCA
Π
actOCP
,
OBP
RANKL
,
is the activator function related to RANKL binding to its receptor
RANK expressed on OCP
D
OBU
,
D
OBP
and
D
OCP
are, respectively, differentiation rates of uncommitted
osteoblast progenitors, osteoblast precursor cells, and preosteoclasts
