Biomedical Engineering Reference
In-Depth Information
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
Porosity
p
FIGURE 5.3
The relationship between
k
F
and
p
in disuse-mode remodeling.
The formula (
c
1
>
c
0
)(
p
/
p
0
)
n
indicates the influence of biological factors.
If this formula vanishes, when Φ converges to zero,
p
will approach 1, which
means that all the bone tissue is resorbed. However, as is well known,
although a mass of bone loss may be observed, bone tissue is not completely
resorbed in the body of a patient who stays in bed for a long period of time.
It is reasonable to predict that there must be some other factors contributing
to bone remodeling as well as the mechanical factor. Qu et al. [2] assume it to
be biological factors that prevent complete bone resorption. The porosity of
the remaining bone tissue is assumed to be
p
0
= 50% and
p
0
≤
p
< 1. The shape
of the curve is defined with
n
= 5 (see Figure 5.3).
It can be seen from Figure 5.3 that as the porosity
p
increases,
k
F
also
increases, which indicates that the bone tissue secretes more growth factors
to deposit more bone material and restrain bone resorption.
This constitutive model is based on first-order, nonhomogeneous, nonlinear
differential equations (5.2), which govern the evolutionary state variables of
porosity and damage. The environmental stimulus Φ is regarded as the forc-
ing function. The rate equation (5.2) involves implicitly the BMU activation
frequency
f
a
,
which itself is not an independent state descriptor, as it is alge-
braically related to
p
and Φ in Equations (5.13) and (5.16). The algorithm is
implemented using a simple forward Euler scheme to integrate Equation
(5.2) with respect to time. The integral in Equation (5.12) is calculated using
the history of the daily average activation frequency. Then it can be used to
analyze the bone modeling and remodeling process. Numerical simulation
follows in the next section.
