Biomedical Engineering Reference
In-Depth Information
Assuming a distribution
, constant at each element, condition (2.18) becomes
local, allowing one to obtain a stationary condition of the type
ρ
2
b
+
b
+
i
∂
C
ρ
∂ρ
ε
(
u
)
·
ε
(
u
)d
∂
C
ρ
∂ρ
ε
(
u
)
·
ε
(
λ
)
d
∂
σ
(
u
)
∂ρ
1
·
ε
(
λ
)
d
b
b
B
e
=
=
1
α
V
b
(2.24)
where
V
b
=
e
i
is the element
(or group of elements) of the interface depending on the density of element
e
.In
practice, this contribution provided by the integral over
b
d
is the volume of element
e
of the bone and
e
i
is only computed for
those elements of the bone that are in contact with a (generally single) element
of the interface. The satisfaction of this optimality condition is sought by a fixed
point scheme that updates the density values according to the following evolution
expression:
e
min
(
e
,
ρ
min
if
B
e
ρ
≤
max
(
e
,
ρ
min
n
n
e
n
1
−
ξ) ρ
1
−
ξ) ρ
e
if max
(
1
−
ξ) ρ
e
,
ρ
min
≤
B
e
ρ
≤
min
(
1
+
ξ) ρ
e
,
ρ
max
B
e
ρ
n
+
1
n
n
n
e
n
ρ
=
max
(
1
+
ξ) ρ
e
,
ρ
max
if min
(
1
+
ξ) ρ
e
,
ρ
max
≤
B
e
ρ
e
n
n
n
e
(2.25)
where
n
refers to the iteration number,
ξ
is a movable limit, and
ς
is a numerical
stabilization parameter. It can be observed that in Eq. (2.25) lateral constraints
over
ρ
(
ρ
∈
[
ρ
min
,
ρ
max
]) were included. In order to obtain a stable sequence of
iterations, the
ξ
and
ς
parameters must be appropriately selected. The variable
φ
defining the orientation of the microstructure is not present in the sensitivity
expressions. A heuristic reorientation following a weighted sum of the principal
strain directions computed at the centroid of each element is used instead [52]:
Nlc
j
ε
i
j
e
i
e
ε
=
1
ω
(2.26)
j
=
Ifthemassconstraintmustbesatisfied,the
parameter must be updated in
Eq. (2.25) at each step. This procedure is used in this study to define the bone
distribution at the beginning of the remodeling process, that is, the initial condition
of the femur before surgical intervention. With this aim, the bone alone is treated,
excluding shaft and interface conditions. The iterations seek a stationary point
satisfying Eq. (2.18) plus lateral constraints. Clearly, this step may be replaced by a
direct experimental definition of
α
ρ
and
φ
by, for example, image processing.
and
φ
are defined, the femoral head is removed, and shaft and
interface regions are included in the model. The remodeling process starts as a
consequence of the stress/strain changes in the system following the evolution law
provided by Eq. (2.18). It must be emphasized, however, that the mass is allowed to
change during this process, which means that the Lagrange multiplier
α
remains
fixed. In the present study, the value of
α
is that obtained at the end of the first
stage in which stationarity conditions for the bone are satisfied.
It should be noted that it is necessary to solve two finite element problems at
each load case of each iteration: the first is to find the solution to the nonlinear
Once the initial
ρ
