Biomedical Engineering Reference
In-Depth Information
5.6.2 A Constitutive Remodeling Model
For internal bone remodeling, the change of porosity is still defined by
F
(5.37)
=−
pU U
R
where
U
R
and
U
F
are, respectively, the total resorbing rate and refilling rate.
They are assumed to be linear functions of the density of resorbing
NN
(
i
,
s
)
and
NN
(
i
,
s
)
refilling as
F
F
i
UkN
=
,
RRR
(5.38)
i
s
UkNN
=
(
+
)
F
F
R
R
where
k
R
and
k
F
are, respectively, the correlation coefficient of bone resorbing and
refilling rates
k
F
is defined by Equation (5.16)
NN
s
are the numbers of resorbing BMUs in internal and surface
bone remodeling
NN
i
and
R
R
i
s
are the numbers of refilling BMUs in internal and surface bone
remodeling, respectively
and
F
F
Similarly to Equation (5.25),
N
i
here is determined by
2
π
R
2
(
)
∫
∫
k
Φ
N
(max)
1
−
e
rdrd
θ
R
R
0
R
N
i
=
(5.39)
1
(
)
R
2
2
π
ba
−
and
N
s
is assumed to be a linear function of the surface velocity
U
e
as
aU
kba
π
π−
2
aU
kb a
2
s
e
e
N
=
=
(5.40)
(
)
(
)
2
2
2
−
2
R
R
where
U
e
is calculated by
(
)
s
k
R
Φ−Φ
Uk
=
1
−
e
(5.41)
L
e
R
or
da
dt
(
)
s
k
RL
Φ−Φ
−= −
k
1
e
(5.42)
with
k
s
being the velocity of surface remodeling when the bone is in the
disuse state.