Biomedical Engineering Reference
In-Depth Information
N
0
=
k
s
p
(5.22)
where
k
s
is the proportional coefficient that denotes the maximum growth
factors consumed by the internal bone remodeling per unit volume. Because
the variation of bone surfaces is very small, the surface bone remodeling rate
can be written as
da
dt
db
dt
(
)
(
)
−= −
kN
N
,
=
kN N
−
(5.23)
e
F
0
p
F
0
where
k
e
is the coefficient of the endosteal remodeling rate and
k
p
is the
coefficient of the periosteal remodeling rate.
Consider a hollow circular cylinder of bone that is subjected to a quasistatic
axial load
P,
an external pressure
P
t
,
an electric load φ
a
(a n d/o r φ
b
), and a
magnetic load ψ
a
(a n d/o r ψ
b
). The related equations for constitutive rela-
tion, equilibrium equation, boundary conditions, and expressions of strains
defined by Equations (3.71), (3.74)−(3.76), (3.6), (3.77), and (3.82)−(3.86) are still
applicable here.
To show the results more clearly, changes of the inner and outer surfaces
are again written in terms of the following nondimensional parameters (see
Equation 4.17):
a
a
b
b
ε= − =−
1,
1
(5.24)
0
0
It is obvious that Equations (3.82)−(3.86) show that all the variables are
dependent on
r
only. Therefore, the environmental stimulus must also be a
function of
r.
As the bone remodels itself, the stimuli of the inner and the
outer surfaces vary in the radial direction, which results in variation of the
BMU activation frequencies and the quantity of growth factors in differ-
ent locations of the bone. However, He et al. [3] assumed that the growth
factors can be uniformly distributed to the bone structures by means of
the fluid flow. The homogenized quantities of the growth factors can be
shown as
2
π
R
2
(
)
∫
∫
1
k
Φ
N
1
−
e
rdrd
θ
r
R
(max)
1
0
R
N
=
(5.25)
1
(
)
R
2
2
π−
ba
where
R
1
denotes the inner radius at which the environmental stimulus
exceeds the remodeling threshold and
R
2
denotes the outer radius at which
the environmental stimulus exceeds the remodeling threshold.
He et al. [3] next simulated the bone surface remodeling process numeri-
cally, based on the preceding model.
