Biomedical Engineering Reference
In-Depth Information
number of osteocytes (long time scale) in remodeling BMUs. Using Equations
(7.34) and (7.39), and based on experimental results [46,66], the concentration
changes of NO and PGE
2
during the bone remodeling process are defined as
N
dNO
dt
∫
=⋅
KR OSTn
⋅
⋅
MS
dN
−⋅
D
O
( 7. 4 0 )
NO
IFSS
OST
NO
0
N
dP
dt
2
∫
=⋅
KR OSTn
⋅
⋅
MS
dN
−⋅
DP
2
( 7. 4 1)
P
2
IFSS
OST
P
2
0
where
K
NO
is the secretion rate of NO by osteocytes
n
is the number of loading bouts per day
K
P2
is the secretion rate of PGE
2
by osteocytes
D
NO
is the rate of degradation of NO
D
P
2
is the rate of degradation of T PGE
2
There are now six unknown variables—
OBP, OBA, OST, OCA, NO,
and
P
2—and six independent equations: (7.6)-(7.9), (7.40), and (7.41). This ordinary
differential equation system can be numerically solved and the numerical
results for each variable can be obtained.
Then, following the method used in the work of Wang et al. [24], Qin and
Wang [2] assumed that bone formation and resorption rates are proportional
to the number of active bone cells—that is,
dBMC
dt
(
)
(
)
(
)
(
)
=⋅
K
BA tOBA t
−
−⋅
KOCAt
−
OCAt
( 7. 4 2)
for
0
res
0
Note that
BMC
is the bone mineral content in percentage and
K
for
and
K
res
are the relative bone formation and resorption rates. The simulation
starts from a so-called steady state in which the values of model variables
remain constant as initial values such as
BMC
(
t
) = 100%,
OBA
(
t
) =
OBA
(0) and
OCA
(
t
) =
OCA
(0). Therefore, the model equations (7.6)-(7.9) become
(
)
T
β
T
β
D
⋅Π
−
D
⋅
OBP
0
⋅Π
=
0
( 7. 4 3)
OBU
actOBU
,
OBP
rep OBP
,
(
)
(
)
T
β
D
⋅
BP
0
⋅Π−
A
⋅
BA
0
=
0
( 7. 4 4)
OBP
rep OBP
,
OBA
T
OBA
·
OBA
(0)−
A
OST
·
OST
(0) = 0
(7.45)
(
)
RL
T
β
D
⋅Π
−
A
⋅
OCA
0
⋅Π
=
0
( 7. 4 6 )
OCPact OCP
,
OCA
actOCA
,
By solving Equations (7.43)-(7.46), the initial values of the model variables in
Table 7.2 can be obtained.
