Biomedical Engineering Reference
In-Depth Information
Table 6.1 Coefficients of Lotz's law
Bone tissue
Direction
a
1
a
2
a
3
a
4
Cortical
Axial
2.065E+03
3.090E+00
7.240E+01
1.880E+00
Transversal
2.314E+03
1.570E+00
3.700E+01
1.510E+00
Trabecular
Axial
1.904E+03
1.640E+00
4.080E+01
1.890E+00
Transversal
1.157E+03
1.780E+00
2.140E+01
1.370E+00
a
2
E
i
¼ a
1
q
app
ð
6
:
4
Þ
a
4
r
i
¼ a
3
q
app
ð
6
:
5
Þ
Being E
i
the elasticity modulus and r
i
the ultimate compression stress in direction
i, both are expressed in MPa and the apparent density q
app
in g/cm
3
. The coefficients
a
j
are presented in Table
6.1
for the axial and transversal direction for both cortical
and trabecular bone.
Thus, besides considering the bone anisotropy, Lotz propose distinct material
laws for the cortical and the trabecular bone.
6.2.2 Proposed Material Law
A recent experimental study by Zioupos [
26
], reinforces the idea that density is a
salient property of bone and plays a crucial role in determining the mechanical
properties of both its trabecular and cortical structural forms. The study, using the
measured apparent density, was able to objectively isolate the bone in trabecular and
compact forms. The results from the work of Zioupos show that the relation between
E
i
and q
app
is not an increasing monotonic function, as it is Lotz law, but instead it is
a 'boomerang'-like pattern. The experimental work of Zioupos also shows that the
law governing the mechanical behaviour of the bone tissue is the same for cortical
bone and trabecular bone. This experiment permits to differentiate the trabecular
bone from the cortical bone based only in the apparent density measure,
trabecular bone
!
q
app
1
:
3 g/cm
3
cortical bone
!
q
app
[ 1
:
3 g/cm
3
ð
6
:
6
Þ
The identified bone minimum and maximum apparent density in Zioupos work
was: q
min
app
¼ 2
:
1 g/cm
3
. In Fig.
6.5
it is presented the exper-
imental data shown in Zioupos work [
26
]. Thus, following the conclusions of
Zioupos [
26
], the author and co-workers [
27
,
28
], proposed a new orthotropic
mathematical law governing the mechanical behaviour of the bone tissue, which
correlates the apparent density with the bone tissue mechanical properties, unifying
app
¼ 0
:
1 g/cm
3
and q
max
