Biomedical Engineering Reference
In-Depth Information
5.5.1 Equations of Growth and Remodeling
Based on the hypothesis presented in Section 5.3, He et al. [3] proposed a
computable model for the process of surface bone remodeling. In that model
they assumed that
N
R
consists of following two parts:
1
0
NNN
R
=+
(5.17)
R
R
where
N
1
is the population of environmentally stimulated BMUs that is
assumed to be a function of the existing state of damage:
1
1
k
Φ
NN
=
(1
−
e
)
(5.18)
r
R
R
(max)
where
N
R
(max)
1
is the maximum number of environmentally stimulated BMUs,
N
R
(max
1
= 0.8 BMU/mm
2
,
k
r
= -1.6, and the environmental stimulus Φ is
defined by Equation (5.14).
For convenience,
N
F
in Equation (5.2) is divided into two major parts:
NN
s
where
N
i
represents the number of refilling BMUs in internal
bone remodeling and
N
s
denotes the number of refilling BMUs in surface
bone remodeling. Then, Equation (5.2) is rewritten as
i
and,
F
F
=
i
pQNQN
RR
−
(5.19)
FF
where
N
i
is determined by the following [64]:
NNN
if
≤
F
F
0
i
N
=
(5.20)
F
NNN
if
>
0
F
0
As in Equation (5.15), the total number of refilling BMUs
N
s
can be found
by multiplying the quantity of resorbed osteocytes by
k
f
:
NkN
F
s
R
(5.21)
f
where
k
f
is defined in Equation (5.15). When bone tissues are overloaded,
k
f
=
c
1
.
The formulation is for the internal bone remodeling discussed in Section 5.4.
As noted, when the porosity of bone structure is reduced to a certain
magnitude, the growth factors exceed the quantities consumed by the inter-
nal bone remodeling and the excess is transported to the surface of the bone
structure, where new bone material is deposited. He et al. [3] defined this
threshold as
N
0
, which is dependent on porosity
p.
Then they assumed a
linear relation between the porosity
p
and the population
N
0
:
