Biomedical Engineering Reference
In-Depth Information
Ta b l e 4
Calculated moduli for 0.00625% initial cells
Volume fraction
0.00
0.12
0.25
0.37
0.5
0.62
0.75
0.87
1.00
E-Modulus
100
169
335
688
1402
2544
4485
6903
10000
ʽ
0.30
0.27
0.25
0.25
0.24
0.25
0.26
0.28
0.30
K-Modulus
83
125
227
452
905
1670
3084
5124
8333
G-Modulus
38
66
133
276
565
1021
1783
2706
3846
Error
0.05
0.21
0.23
0.23
0.16
0.08
0.03
Ta b l e 5
Calculated moduli for 0.2% initial cells
Volume fraction
0
0.12
0.25
0.37
0.5
0.62
0.75
0.87
1
E-Modulus
100
189
442
943
1897
3179
5052
7215
10000
ʽ
0.30
0.27
0.24
0.23
0.23
0.24
0.26
0.28
0.30
K-Modulus
83
135
284
587
1190
2067
3503
5404
8333
G-Modulus
38
74
178
383
769
1278
2005
2824
3846
Error
0.02
0.04
0.05
0.04
0.03
0.02
0.01
Ta b l e 6
Calculated moduli for 6.4% initial cells
Volume fraction
0
0.12
0.25
0.37
0.5
0.62
0.75
0.87
1
E-Modulus
100
182
483
1116
2226
3608
5455
7462
10000
ʽ
0.30
0.26
0.23
0.23
0.23
0.25
0.26
0.28
0.30
K-Modulus
83
128
301
683
1399
2375
3838
5653
8333
G-Modulus
38
72
196
454
901
1447
2159
2915
3846
Error
0.02
0.04
0.03
0.02
0.02
0.02
0.01
E
2
e
bc
1
E
2
e
b
c
1
E
=
+
(
E
1
−
)
(21)
E
2
and
c
1
is the volume fraction. The shape of the curve is modeled with help of the structural
parameter
b
.
Contrary to a volume fraction variation, three types of volume fractions are chosen
to be constant (at 6, 12 and 25%, respectively) and the percentage amount of initial
cells are varied. The plots are shown in Fig.
11
. The stiffness increases clearly towards
a maximum with increasing dispersion. The stiffness declines again behind this
maximum. This maximum, thereby, shifts with increasing volume fraction.
Equation (
21
) can be used to fit real measurements. Figure
12
shows the measure-
ments of Ashmann et al. [
1
,
2
] for bovine cancellous bone taken from femur and Rho
et al. [
3
] for human cancellous bone taken from different bone types. The spreading
of the measurements can be evolved completely, only by varying the structure para-
meter b. This means, the spreading can be explained by structure since both moduli
are retained constant.
E
1
and
E
2
are the respective elasticity moduli of both materials with
E
1
>
