Biomedical Engineering Reference
In-Depth Information
equilibrium problem (Eq. 2.27) and the second is the solution to the adjoint linear
problem (Eq. 2.17).
2.3.4
Model for Interfacial Adaptation
It is considered here that the mechanical environment is the main aspect regulating
the tissue differentiation and that biological factors are directly related to a mechan-
ical stimulus. The one selected is related to the relative micromovements between
the prosthesis and the bone, which in this case correspond to the shear strains
of the gasket elements. This model is based on a simple mixture rule that allows
the coexistence of the bone and fibrous tissue at the same interfacial location. Two
relative quantities
α
α
f
quantify the occurrence of bone ingrowth and fibrous
tissue respectively, satisfying the condition:
b
and
=
1 (2.27)
The internal forces
F
int
of each gasket element are computed from the contribution
of each material, as follows:
α
+
α
b
f
F
int
(2.28)
where
F
b
and
F
f
are the corresponding internal forces of each material to the same
strain. The evolution of quantity
α
b
(
α
f
is related to
α
b
by Eq. (2.27)) is proportional
to the difference between the current state and the state that corresponds to the
current strain (displacements):
d
α
b
d
t
=
υ
b
F
b
+
α
f
F
f
α
=
[
M
(
u
)
−
α
b
]
(2.29)
where
M
is denominated the stimulus for the osteogenesis calculated from the
current normal and shear strain of the interfacial element. The parameter
υ
is
related to the evolution rate and was introduced in such a way as to limit the
changes between each time-step. The stimulus
M
for the osteogenesis is based on
the tissue differentiation theory from Carter
et al
. [38]. The simplified graphical
representation of this theory can be seen in Figure 2.7, where the value of the
stimulus
M
depending of the normal and tangential relative displacements
u
n
and
u
t
is shown. Interfacial elements that undergo relative displacements within the
limits
u
t
u
n
max
, 0] will contribute to the formation of
interfacial bone, whereas the elements submitted to displacements outside these
limits tend toward the formation of the fibrous tissue. A transition zone is also
incorporated in order to allow for a smooth change between the two extreme states.
Thevaluesassumedfortheselimitsare
u
t
max
150 µmand
u
n
max
6 µm.
The updating of the relative coefficients follows a simple explicit time integration
scheme:
α
∈
[
−
u
t
max
,
u
t
max
]and
u
n
∈
[
−
k
+
1
k
b
k
+
1
k
+
1
=
α
+
α
t
,
α
=
1
−
α
(2.30)
b
b
f
b
The evolution law presented allows that, during the entire course of the remodeling
simulation, the interfacial elements can evolve toward the formation of fibrous
