Biomedical Engineering Reference
In-Depth Information
tissue at a specified region is estimated using a forward Euler method, and then obtained
by Equation 20.18:
ρρ
ρ
d
dt
m
=+
∆
t
(20.18)
i
+
1
i
If damage accumulation
ω
was above the critical value, a random number generator was
used to determine which element becomes a remodeling site according to BMU activa-
tion probability, which depends on the local damage state. Once an element is activated, it
begins a microdamage-remodeling program; otherwise, it will return back to the adaptive
remodeling program. Therefore, when undergoing remodeling, the rate of change of local-
ized bone density is given by
fort
∈+
∈+ ++
∈+ ++++
[
tt
,
T
]
ρ
(,)
xt
R
R
dxt
dt
ρ
(,)
(20.19)
r
fort
(
(
tTtT T
,
]
=
0
R
R
O
ρ
(,)
xt
fort
tT Tt TTT
,
]
F
RO
ROF
where
ρ
r
is the bone density experiencing remodeling,
T
R
is resorption period,
T
O
is rever-
sal period, and
T
F
is formation period.
ρ
.
R
represents the rate of osteoclastic resorption and
ρ
.
F
represents the rate of osteoblastic formation. Therefore, at the different phases of bone
remodeling, bone density is determined by
tT
R
+
ρ
(,
xt
+=
T
)
ρ
( ,)
x t
−
ρ
(,)
x tdt
∫
r
R
r
R
t
ρ
(,
xt
++ =
TT
)
ρ
( ,
xt
+
T
)
(20.20)
ρ
(,)
xt
=
r
R
O
r
R
r
tT
R
+++
T
O
T
F
ρ
(,
xt
+++=
TTT
)
ρ
( ,
x tT T
++ +
)
ρ
(,)
x tdt
∫
r
R
OF
r
R
O
F
tT
R
++
T
O
The combination of adaptive remodeling and microdamage remodeling algorithms changes
the localized bone density. As illustrated in Equations 20.9 and 20.10, the elastic modulus
depends on the density value, elastic modulus per element is updated to calculate new
stress/strain field using FEA, and then a new loop is started. The process is repeated time
after time to simulate the evolution of trabeculae configuration.
20.3
aPPlICatIonS In tooth movement and dental ImPlantatIon
20.3.1 S
imulation
of
o
rtHodontic
t
ootH
m
oVement
Figure 20.5 presents the final simulation results of apparent bone density distribution. The model
cross-section reveals the effects of four different orthodontic loadings. For case (a), under the action
of the tipping torque, a buccal tipping around the center of resistance was achieved at the end
of simulation. As indicated in Figure 20.5a, the cancellous bone density increased near the tooth
and formed a high-density region connecting the two sides of the cortical bone. For case (b), the
surrounding bone density increased dramatically to support the torque loading around the
z
-axis
(Figure 20.5b). For case (c), besides a buccal tipping torque, a force along the
y
-axis was applied to
achieve bodily tooth movement in the coronal direction (Figure 20.5c). The pattern of bone density
distribution was similar to case (a); however, because the tooth was extruded from the alveolar bone,
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