Biomedical Engineering Reference
In-Depth Information
3.
Trabeculae remodeling algorithm with both adaptive and microdamage remodeling
mechanisms
. Initially, damage rate
ω
.
was defined to assess the state of damage. If damage
accumulation
ω
= 1 was a failure, and if the rate of damage accumulation was assumed
to be linear, then Miner's rule was employed to define the damage rate (McNamara and
Prendergast 2007):
ω
.
=
1
(20.11)
N
f
where
N
f
denotes the number of cycles required for a failure of the material at a given
stress. In the case of bones, the empirical equation of Carter et al. (1976) was used to cal-
culate
N
f
:
log
NH
=
log
σ ++ +
JT
KM
ρ
(20.12)
i
i
f
where
σ
i
is the stress (MPa),
T
is the temperature (°C), and
H, J, K
, and
M
are all empirical
constants. Then, the damage accumulates as
ω
.
t
=
dt
(20.13)
where
ω
denotes the state of local damaged bone tissue and
t
the duration of calculation.
When the damage accumulation
ω
was below a certain damage threshold, the adaptive
remodeling program was used. Based on previous research (Ruimerman et al. 2005;
Xinghua, He, and Bingzhao 2005), strain energy density at location
i
was given by
εε
ρ
i
E
i
i
U
=
(20.14)
i
2
i
where
E
i
is the elastic modulus (MPa) at location
i
(assumed isotropic),
ε
i
is the strain,
and the unit of
ρ
i
is g/cm
3
. At the given location, the stimulus for an element is
S
i
, and is
defined as
SUU
i
=−
i
ref
(20.15)
where
U
ref
represents the reference stimulus. The total stimulus received by an osteocyte is
a sum of stimuli from other osteocytes surrounding it, hence
dxt
dt
ρ
(,)
N
∑
m
=
Bf
i
() ()
xS x
i
0
<≤
ρρ
(20.16)
max
i
=
1
where
ρ
m
is the bone density mediated by adaptive remodeling. Parameter
B
denotes the
proportion coefficient,
N
is the number of sensor cells that contribute to the total stimuli,
and
ρ
max
signifies the maximal bone density.
f
i
(
x
) is a spatial influence function, describing
the decay in signal intensity relative to distance
d
and decay parameter
D
, according to
=
−
dxxD
(,')
f
i
()
xe
(20.17)
where
d
(
x
,
x
') is the distance from osteocyte
x
to surface location
x
' and
D
is the distance
from an osteocyte location where its effect has been reduced to
e
-1
. The density of bone
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