Biomedical Engineering Reference
In-Depth Information
10.6.4
Multicriteria Optimal Geometries -
f
mc
The geometric characteristics obtained for the single objective functions are, in
many cases, opposite. However, to improve uncemented hip prostheses it is
necessary to reduce, at the same time, relative tangential displacement, normal
contact stress, and stress shielding effect. Therefore, a multicriteria optimization
procedure allows us to simultaneously minimize the three single objectives.
With a multicriteria optimization algorithm it is possible to obtain a set of non-
dominated points [44]. In this case four nondominated points were computed for
these weighting coefficients of Eq. (10.7): (
3), (0.6,0.2,0.2),
(0.2,0.6,0.2), (0.2,0.2,0.6). In Table 10.6, the function values for these nondominated
points are presented and they are compared with the results obtained with single
cost functions, that is, (
β
d
,
β
t
,
β
r
)
=
(1
/
3,1
/
3,1
/
(1,0,0), (0,1,0), (0,0,1). It is possible to conclude
that totally coated stems have lower
f
d
and
f
t
function values, but for remodeling
function
f
r
, half coated implants have the lower value. Almost all nondominated
points have better values for the three objective functions compared to the initial
shape, which is not obtained with the single cost functions.
Figures 10.15 and 10.16 show the nondominated shapes for totally and half
coated stems, respectively. When all objectives have the same weight, that is,
β
d
=
β
t
=
β
r
=
1
/
3, a small and circular stem tip is observed in order to reduce
maximum contact stress and also to improve proximal load transfer. On the other
hand, section 2 is larger (parameter
a
) to increase rotational stability. Section 3 is
also larger to improve elbow cortical fixation. Section 4 is thicker to increase contact
surface and almost rectangular to give more rotational stability to the implant.
When the weight for the displacement objective is bigger (
β
d
,
β
t
,
β
r
)
=
6), more
geometric characteristics to minimize interfacial displacement are observed. For
instance, section 1 is large enough, having a slight fixation on cortical bone, section
2 is large and rectangular, and section 4 is also rectangular. However, section 4 is
thicker to reduce contact stress in the calcar region.
β
=
0
.
d
Table 10.6
Objective function values. Minimization of multicriteria function
f
mc
.
(
β
d
,
β
t
,
β
r
)
Totally coated
Half coated
f
d
f
t
f
r
f
d
f
t
f
r
(1,0,0)
21.58
17.84
67.97
26.98
43.95
56.48
(0,1,0)
61.30
8.31
48.13
100.6
9.25
46.23
(0,0,1)
37.69
186.04
17.93
40.21
185.49
17.93
(1/3,1/3,1/3)
27.41
12.64
48.08
32.98
16.23
40.79
(0.6,0.2,0.2)
22.40
15.85
58.54
29.51
20.16
47.94
(0.2,0.6,0.2)
29.37
10.69
48.03
35.70
13.00
42.13
(0.2,0.2,0.6)
32.85
16.21
38.62
38.97
17.12
35.48
Initial shape
40.18
24.26
50.34
52.41
39.03
49.61
