Biomedical Engineering Reference
In-Depth Information
where
K
to
is the relative rate of bone turnover. By substituting Equation (7.42)
into Equation (7.47), it can be observed that BMC contributes to BFE:
dBFE
dt
dBMC
dt
(
)
(
)
=
+
K
⋅
BA tOCA t
+
( 7. 4 8)
to
Qin and Wang [2] proposed that BMC linearly contributes to BFE, which is
the first term in Equation (7.48). The second term in Equation (7.48) represents
the contribution of bone turnover (
OBA
(
t
) +
OCA
(
t
)) to BFE, in order to con-
sider the structural effects of optimized collagen alignment on bone strength
in the newly formed bone. It has been observed in experiments [6,67,69] that
the contribution of bone turnover to the increment of bone strength diminishes
as bone turnover increases. Qin and Wang postulated that the square root of
bone turnover contributes to the derivative of BFE with respect to time. By
applying the superposition principle, the relationship of BFE with BMC and
bone turnover can be obtained in the form of Equation (7.48). The validity of the
proposed equations is tested in the following section. This equation might not
be completely accurate, but it fits the model well, at least providing an alterna-
tive understanding of the relationship between BFE, BMC, and bone turnover.
In the work reported in Qin and Wang [2], all the parameter values used in
the model are from experiments, and they are not amended or modified to
fit the experimental data. A cursory examination of the parameters indicates
two classes. The first class corresponds to the physicochemical parameters:
K
D
1,
T
β
,
K
D
2,
T
β
, K
D
3,
T
β
, K
D
4,
PTH
, K
D
5,
PTH
, K
D
6,
RL
, K
D
7,
NO
, K
D
8,
NO
, K
D
9,
P
2
, T
acc
,
K
A
1,
RL
,
K
A
2,
RL
,
k
NO
,
k
PTH
,
k
P
2
,
k
T
β
, β
PTH
, β
RL
, β
OPG
,
D
PTH
,
D
RL
,
D
OPG
,
D
T
,
D
NO
,
D
P
2
and τ.
These parameters generally remain fixed under different physiological con-
ditions. They are easily measured through experiments and have values
reported in the literature.
The second class of parameter value may fluctuate slightly if the larger
physiological environment changes. They are
A
OBA
,
D
OBU
,
D
OBP
,
T
OBA
,
A
OST
,
D
OCP
,
A
OCA
,
K
NO
,
K
P
2
,
K
for
,
K
res
,
K
to
,
RK,
α,
OPG
max
, and
R
RL
.
The values for these
parameters are averaged from a range of acceptable values for each by
checking the literature. In the analysis presented in Qin and Wang [2], most
parameter values are referred from previous models in references 3 and 23.
Values of the new parameter used in this chapter are from references 20, 21,
56, and 57 (see Table 7.1).
β
7.3 Numerical Investigation
Mathematical models of biology are a form of complex hypothesis. To test
the validity of the hypothesis, external data are utilized to see if the model
matches the experiments. Ideally, the model should be tested by as many
