Biomedical Engineering Reference
In-Depth Information
a
a
b
b
p
2
=
p
=
4.5
Figure 10.3
Influence of parameter p in Eq. (10.2),
representation of first quadrant with
a
=
b
. (Adapted from
Ruben
et al
. [2, 3].)
10.4.1
Design Variables and Geometry
To define the design variables, let us consider the stem transversal sections
defined by,
x
a
p
y
b
p
+
=
1
(10.2)
in a local coordinates system
xy
. The parameters
a
,
b
,and
p
characterize the
section geometry as shown in Figure 10.3. If
a
b
, then the section is circular
for
p
=
2andbecomesasquarewhen
p
increases. For
a
=
b
,circularandsquare
sections become elliptical and rectangular, respectively. The 14 design variables (
d
)
defining the four key sections are presented in Figure 10.4. These design variables
are the parameters
a
,
b
,and
p
that are defined in Eq. (10.2), and an appropriate
interpolation of the key sections is used to obtain the stem geometry. The initial
geometry, observed in Figure 10.4, is based on the commercial Tri-Lock prosthesis
from DePuy, a Muller-type stem with a high survival rate.
All 14 design variables have lower and upper bounds to maintain the prosthesis
inside the bone. Additionally, 10 linear constraints are considered to achieve
clinically admissible stem shapes:
=
h
1
=
d
1
−
d
4
≤
0;
h
2
=
d
2
−
d
5
≤
0;
h
3
=
d
4
−
d
7
+
c
3
≤
0
h
4
=
d
4
−
d
8
≤
0;
h
5
=
d
5
−
d
9
≤
0;
h
6
=
d
7
−
d
11
+
c
6
≤
0
=
−
≤
=
−
≤
h
7
d
12
d
8
0;
h
8
d
9
d
13
0
d
11
+
d
4
h
9
=
d
11
−
d
7
+
c
9
≤
0;
h
10
=
d
7
−
≤
0
(10.3)
2
Constraints
h
3
,
h
6
,and
h
9
are necessary to always obtain b-splines with negative
slope, as illustrated in Figure 10.5. The values for
c
3
,
c
6
,and
c
9
depends on geometry
and in this case are
c
3
=
c
6
=
3
.
5, and
c
9
=−
9. Finally, constraint
h
10
assures a
convex b-spline, as shown also in Figure 10.5.
Without these constraints, the optimization process can lead to clinically unfea-
sible stem shapes that, at least, imply special insertion techniques or make the
removal process difficult [13].
