Biomedical Engineering Reference
In-Depth Information
for stem design is to minimize the difference between strain energy density before
and after the implant (see for instance Chang
et al
. [19]). However, in this chapter
another criterion is used, with the objective to minimize the bone loss in proximal
femur. Since after surgery, the stiff stem supports most of the load, the strain
energy in bone decreases too much. Thus, the objective is to increase the strain
energy density in proximal femur. To reach that goal the remodeling function
f
r
was defined by,
D
P
1
U
j
f
r
=
(10.6)
NC
P
N
bp
j
α
=
1
=
P
where
N
bp
is the number of elements in proximal femur, that is, where the
trabecular and cortical bone exists,
U
j
represents the strain energy density, and
P
istheloadcase.Theconstant
D
is taken equal to 10
3
.
10.4.5
Multicriteria Objective Function
The three single cost functions are important to understand the influence of
the stem shape on the remodeling and interface displacement and stress is in
an individual manner. However, a multicriteria objective function is necessary
to obtain simultaneously the implant geometries with less stress shielding and
improved stability, that is, with better performances. To do that, a multicriteria
objective function combining the three single cost functions was considered,
f
d
f
d
−
f
t
−
f
0
t
f
r
−
f
r
f
r
f
mc
=
β
f
d
+
β
f
0
t
+
β
(10.7)
d
t
r
f
d
−
f
i
t
−
−
f
r
where
f
d
,
f
t
,and
f
r
and
f
d
,
f
t
,and
f
r
are the minimums and the initial values
of
f
d
,
f
t
,and
f
r
, respectively; and
β
d
,
β
t
,and
β
r
are the weighting coefficients with
β
d
+
β
t
+
β
r
=
1.
The multicriteria cost function is based on a weighting objective method as
described in Osyczka [36], and Marler and Arora [37] reported that this approach to
normalize objective functions is the most robust.
10.5
Computational Model
The problem described above is solved numerically using a suitable discretization
by finite elements and appropriate optimization methods. Next, the optimization
algorithm is described in detail and the finite element mesh is presented.
