Biomedical Engineering Reference
In-Depth Information
Thus, Equations (4.2) and (4.3) can be written as
2
da
dt
b
ba N
1
1
1
e
e
e
e
e
−=
N
+
+
N
ba N
+
C
0
(4.15)
1
2
3
4
2
2
b
a
2
2
b
a
ln
a
ln
2
2
db
dt
b
ba N
a
ba N
1
1
1
p
p
p
p
p
p
=
N
+
+
+
N
ba N
+
C
0 (4.16)
1
1
2
3
4
2
2
2
2
b
a
2
2
b
a
ln
b
ln
where
=−′
CCN
p
p
3 .
0
0
4.2.3 Approximation for Small Changes in Radii
It is apparent that Equations (4.15) and (4.16) are nonlinear and cannot, in
general, be solved analytically. However, the equations can be approximately
linearized when they are applied to solve problems with small changes in
radii. In the bone surface remodeling process, Qin, Qu, and Ye [1] assumed
that the radii of the inner and outer surface of the bone change very little
compared to their original values, as was done in Section 2.3.2. This means
that the changes in a ( t ) and b ( t ) are small. This is a reasonable assumption
from the viewpoint of the physics of the problem. To introduce the approxi-
mation, two nondimensional parameters [3,4],
a
a
b
b
ε= − =−
1,
1,
(4.17)
0
0
are adopted in the following calculations. As a result, a ( t ) and b ( t ) can be
written as
a ( t ) = (1 +ε ( t )) a 0 , b ( t ) = (1 + η ( t )) b 0 , (ε, η << 1)
(4.18)
Since both ε and η are far smaller than one, their squares can be ignored
from the equations. Consequently, we can have the following approximations:
2
2
ba LL a
b
2
≅+ ε−η
2
(
)
(4.19)
0
0
2
2
2
b
0
2
2
ba LL b
a
2
0
≅ ′ +
2
(
ε −η
)
(4.20)
0
0
2
2
2
a
0
1
2
2
2
ba LLa
≅+ ε− η
2(
b
)
(4.21)
2
2
0
0
2
2
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