Journal of Medical and Biological Engineering, 26(1): 1-7
1
Effects of Bone Mineral Fraction and Volume Fraction on
the Mechanical Properties of Cortical Bone
James Shih-Shyn Wu
Hsiao-Che Lin
Jui-Pin Hung1
Jian-Horng Chen2,*
Institute of Mechanical Engineering, National Chung-Hsing University, Taichung, Taiwan, 402 ROC
Department of Mechanical Engineering, National Chin-Yi Institute of Technology, Taichung, Taiwan, 401 ROC
2
Department of Physical Therapy, Chung Shan Medical University, Taichung, Taiwan, 402 ROC
1
Received 29 Aug 2005; Accepted 7 Nov 2005
Abstract
This study evaluated how bone mineral fraction and volume fraction influence bovine cortical bone strength. Dual
energy X-ray absorptiometry (DEXA) was applied to determine the mineral content of each bovine cortical specimen.
The water displacement method was applied to measure pore volume and, accordingly, calculate porosity and bone
mineral fraction. Additionally, the mechanical properties of specimens were obtained with a material test system (MTS).
This study derives three two-parameter power law functions for Young’s modulus, toughness and the ultimate strength
of wet cortical bone. Analytical results indicate that the change in volume fraction exerts a stronger influence on the
biomechanical properties of the cortical bone than on those of the bone mineral fraction. Results of this study provide a
valuable reference for biomechanical research or as reference data for clinical diagnosis.
Keywords: Bone mineral fraction, Bone volume fraction, Porosity, Biomechanical property, Young's modulus
Introduction
Osteoporosis is a serious public health problem for
women in midlife, and progresses with age. Osteoporosis
symptoms include both low bone mass and microarchitectural
changes in bone tissue, which raise susceptibility to bone
fractures from minor traumas [1]. Reduced bone mineral
density (BMD) appears to be a major determinant of bone
fragility [2-4]. Assessment of cortical bone according to
mechanical properties is relevant for predicting fracture risk
and the selection of suitable therapeutic strategies for
orthopaedic surgery or rehabilitation [5-7].
Relevant literature indicates that the mechanical
properties of bones are primarily determined by their mineral
content [8-10]. Carter and Hayes showed that the Young’s
modulus of trabecular and cortical bone were directly
proportional to the cube of the apparent wet density [11].
Schaffler and Burr discovered that Young’s modulus of bovine
cortical bone is directly proportional to the apparent density
raised to the power of 7.4 [12]. Keller et al. reported that the
bending Young's modulus of human compact bone is directly
proportional to the dry apparent density raised to the power of
1.54 [13]. Wachter et al. [14] in 2002 indicated strong
correlation between cortical bone mineral density and
mechanical strength.
Not only were the strength and stiffness of cortical bone
* Corresponding author: Jian-Horng Chen
Tel: +886-4-24730022 ext. 11764; Fax: + 886-4-23248176
E-mail: jhchen@csmu.edu.tw
proportional to the bone mineral density [15-17], but the
porosity was also noted as an important factor influencing
skeleton strength [18, 19]. The cortical structure contains
Haversian canals, vascular canals and lacunae, which are
generalized as pores or cavities of the cortex. Porosity (P) is
defined as pore volume ( V P ) per unit whole/total volume ( VT )
[20]. Schaffler found that the elastic modulus of bovine
cortical bone fell as the power (-0.55) of porosity increased in
the tensile test [12]. Carter and Hayes in 1976 [21] derived the
equation of the Young’s modulus, E = kV f3 , from cancellous
bone and compact bone, where V f represents the volume
fraction. Currey [22] obtained a strong relationship between
Young’s modulus and both calcium content and volume
fraction. Bell et al. [23] pointed out that increased porosity and
higher prevalence of giant canals both have a significantly
negative influence on the ability of the cortical shell to
withstand stress.
As mentioned above, previous studies discussed the bone
mineral content or porosity with Young’s modulus either
individually or combined as bone density. However,
Hernandez et al. [24] found that the bone volume fraction (i.e.
1 minus the porosity) and mineral fraction were poorly
correlated ( r 2 = 0.01 ), suggesting the two are independent
parameters. The variation of bone mineral fraction and bone
volume fraction may differ during bone changes. Additionally,
the bone mineral fraction in bone can be considered to
represent “bone quality”, while bone volume represents “bone
quantity”, which are two independent variables. This study
J. Med. Biol. Eng., Vol. 26. No. 1 2006
2
Femoral Shaft Axis
Compression
Plate
Screw for Fastening
on Upper Crosshead
Cortical Specimen
Basin
Midline
Screw for Fastening on
Machine Platform
Cortical Specimen
Medial
Lateral
Figure 1. Cortical specimen of bovine femur.
thus evaluates how bone volume fraction and bone mineral
fraction affect bovine cortical bone strength. Three
two-parameter power law functions for the mechanical
properties of wet cortical bone were derived.
Materials and Methods
Specimen preparation
Twenty-nine 2–3-year-old bovine femoral bones were
adopted in this study. The femoral diaphysis of each bone was
extracted and the fibrous periosteum removed. It was then
maintained at −20 °C for further manipulation. Only one
cylindrical specimen was extracted from each femur at the
medial part, and its axis was parallel to the femoral shaft axis
(Fig. 1). Dong and Guo [25] in 2004 proved that longitudinal
Young’s modulus has a significantly negative relationship with
cortical bone porosity, while no such relationship exists
between transverse Young’s modulus and porosity. Since the
thickness of the cortical shell was capricious among different
femurs, specimens were machined to a diameter of
7.28 ± 0.80 mm and a length of 12.50 ± 0.21 mm by a
water-cooled diamond lathe and a milling machine. The bone
specimens were irrigated during manipulation with ringer’s
lactate to prevent drying and overheating [13]. The total
volume VT of each specimen can be computed from
1
VT = D 2πL , where D and L represent the diameter and
4
length, respectively.
Measurement of bone mineral content
DEXA scans were performed on each specimen with the
EXPERT-XL system (LUNAR) to obtain bone mineral content
( WM ). To simulate an in vivo setting, each specimen was
immersed in water to a depth of 10 cm [26], which has been
shown to accurately simulate the surrounding soft tissue [27].
To investigate whether the bone mineral content relates to the
scan direction, a longer bone slice was chosen, and applied the
x-ray beam parallel followed by the perpendicular to the
femoral shaft axis. This simple experiment proves that the
measured bone mineral content is unrelated to the scan
direction.
Figure 2. Compression apparatus.
Measurement of dry and wet weights
All cortical specimens were thoroughly defatted in
full-strength chloroform [28] and placed in an incubator (YIH
DER LE-549) at 50 °C for 72 hours to dry to a constant weight
(<0.05% change) to remove excess fluid. The moisture of the
specimen was thoroughly evaporated to leave only the solid
frame. As each specimen was taken from the incubator, its dry
weight ( WD , solid frame weight) was immediately measured
by an analytical balance (Mettler AE240-S), which can
measure weights to 0.0001 g. The bone water content of slice
was assumed to be 0 g at this point. According to the porous
media theory [29], porous structures are composed of a ‘solid
skeleton’ and ‘interstitial fluid’, in which the voids of the solid
skeleton are saturated with fluid [30]. Thus, if specimens are
sufficiently soaked, the fluid content can be used to estimate
the void volume. Pilot studies performed in our laboratory
found that the specimen weight was almost unchanged after 2
days (2880 min) of soaking. All specimens were soaked with
distilled water for 2.5 days (3600 min) to reach at least 99%
saturation, where 99% of the cortical pores are occupied by
fluid. The surface water was then gently wiped off the
specimens, which were then weighed ( WW ).
Calculation of bone volume fraction and mineral fraction
For each saturated specimen, the void spaces were filled
with W f g of distilled water. The void volume ( VP ) was
then calculated by the formula VP = W f 0.9971 , where
0.9971 is the density of distilled water, and W f = WW − WD .
The porosity P was calculated by the formula P = V P VT ,
and the bone volume fraction α = 1 − P . Dividing mineral
content WM by the solid bone mass WD yields the bone
mineral fraction γ .
Measurement of the Young’s modulus, Ultimate Stress and
Toughness
The mechanical tests were performed by the Chung Shan
Institute of Science and Technology, Ministry of Defense,
Taiwan. A custom-made compression apparatus was adopted
(Fig. 2), which included a basin and a compression plate. The
basin with a screw at the bottom center and the compression
plate with a screw at the upper center were machined from a
round iron bar. Both components were formed in one piece and
fastened onto the apparatus platform with a screw thread to
prevent any error resulting from gap and buckling. The cortical
Mechanical Properties of Bone
3
specimen was erected at the center of the basin. The centers of
the specimen, basin and compression plate were all aligned
with the central screw axis. The specimens were loaded at a
crosshead rate of 0.5mm/min to minimize non-linear effects.
The compression test results were plotted as stress-strain
curves and applied to calculate the mechanical compression
properties.
Numerical analysis
Assume that the two-parameter function is given by:
Y = Kα a γ b
(1)
where K, a and b represent undetermined variables; α
represents the bone volume fraction, γ represents the
mineral fraction, and Y represents either Young’s modulus,
ultimate strength, or toughness. Taking the logarithms for Eq.
(1) at two sides yields:
ln Y = ln K + a ln α + b ln γ
(2)
Figure 3. Stress-strain curve of the compression test, indicating
how Young’s modulus, ultimate stress, and toughness is
obtained.
Table 1. Experimental data of the twenty nine cortical specimens.
Given ln Y = Y , ln K = K , ln α = α , and ln γ = γ ,
Eq. (2) could be expressed as:
Y = K + aα + bγ
(3)
Given a set of experimental data with (α i , γ i , Yi ) , i = 1 to
N, the data were first modified to (α i , γ i , Yi ) by taking the
logarithms. Assuming that the predicted value Yi ′ satisfies
Yi ′ = K + aα i + bγ i , the following method for selecting
( K , a, b) was proposed to minimize
e ( K , a , b) =
1
N
1
=
N
Young’s
Modulus
E (GPa)
0.846
0.882
0.866
0.010
0.611
0.685
0.650
0.023
10.10
15.55
12.59
1.26
N
∑ ( K + aα i + bγ i − Yi ) 2
i =1
(5)
We have:
∂e
= ∑ 2(Yi − K − aα i − bγ i )(−1) = 0
∂K
(6)
∂e
= ∑ 2(Yi − K − aα i − bγ i )(−α i ) = 0
∂a
(7)
∂e
= ∑ 2(Yi − K − aα i − bγ i )(−γ i ) = 0
∂b
(8)
Arranging Eq. (6) to Eq. (8) yields:
∑α i
∑γ i 

2
∑ α i ∑ α iγ i 
2 
∑ α iγ i ∑ γ i 
Ultimate
Stress
σ ult
Toughness
(MPa)
(MPa)
140.11
185.05
161.90
16.84
0.97
1.51
1.22
0.16
* Standard Deviation
Results
(4)
i =1
∂ ∂ ∂
,
, ) e (K , a , b ) = 0
∂ K ∂a ∂b
 K   ∑1
  
 a  = ∑ α i
 b  ∑γ
i
  
Mineral
Fraction
γ (-)
N
∑ (Yi ′ − Yi ) 2
where e( K , a, b) denotes a function with three variables
K , a and b, and its minimum occurs at ( K , a, b) , as:
(
Minimum
Maximum
Mean
S.D.*
Volume
Fraction
α (-)
−1
 ∑ Yi 


∑ Yiα i 
 ∑Y γ 
 i i
(9)
Equation (9) could be solved directly, yielding the
variables a, b, and K = e K .
Figure 3 shows the stress-strain curve of compression test.
All specimens show similar tending graphs, and all have a long
straight portion in the curves. The slope of the straight portion
in the curve is the Young’s modulus E (GPa). The toughness is
obtained by computing the area underneath the stress-strain
curve from zero stress to peak stress. Since the strain is
dimensionless, the unit for toughness is given by MPa.
Toughness measures the amount of energy that a sample can
absorb before breaking. Additionally, the ultimate stress σ ult
(MPa) denotes the maximum stress before breaking. Table 1
shows 29 experimental data sets obtained from several
measurements and compression tests. The Young’s modulus,
ultimate stress and toughness values outside the range of
Mean ± 2SD were excluded, and the remainders were then
substituted into Eq. (9), yielding three equations:
E = 20.80α 2.55γ 0.33
(10)
σ ult = 221.72α 3.70 γ −0.50
(11)
Toughness = 1.20α 1.95γ −0.68
(12)
Using Eq. (10)-(12), Table 2 shows the changes of value
for E, σ ult and the toughness as the value for bone volume
fraction α and bone mineral fraction γ fluctuates. The
proportions of mechanical properties change when the values
of α and γ increase or decrease by 5%.
J. Med. Biol. Eng., Vol. 26. No. 1 2006
4
Table 2. Changes of value for E, σ ult and toughness as the value
for α and γ fluctuate
Change
(%)
Volume Fraction α
+5 %
-5 %
Mineral Fraction γ
+5 %
-5 %
E
+13.25 % -12.26 %
+1.62 %
-1.68 %
σ ult
+19.78 % -17.29 %
-2.41 %
+2.60 %
-3.26 %
+3.55 %
Toughness +9.98 %
-9.52 %
(a)
(b)
(c)
Figure 4. Plots of curved surface equations: (a) Young’s modulus E,
(b) Ultimate stress σ ult , (c) Toughness.
Discussion
Bone is a composite material consisting of an organic
phase synthesized by osteoblasts and an inorganic phase
composed primarily of calcium phosphate crystallized as a
nonstoichiometric apatite. The poroelastic solid theory,
proposed by Biot [29], is a solid-liquid biphasic structure
modeled as a deformable porous solid matrix filled with
saturated fluid. The porosity is defined as the percentage of the
fluid volume to bulk volume.
Wang and Feng [31] in 2005 applied composite material
theory to show that the mineral phase is designed to sustain the
forces on the bones, and that it contributes to the bone’s
mechanical properties such as stiffness and strength.
Decreasing the bone mineral content not only reduces the bone
stiffness, but also raises the risk of bone fracture from falling.
Minerals account for 60% to 70% of the dry weight of normal
human bone [32]. The mineral fraction ranges from 0 (osteoid)
to 0.7 (fully mineralized bone), which is calculated as the
mineral content divided by dry weight [24]. Specimens were
obtained for this study from bovine cortical bones, with a
volume fraction α (defined as the bone solid volume divided
by the total volume) ranging from approximately 84.6% to
88.2%, and the bone mineral fraction γ from 61.1% to 68.5%
(Table 1).
Both bone volume fraction and bone mineral fraction
significantly affect the bone’s biomechanical properties [8-12,
14, 23], and neither should be neglected. This study differs
from others in that it applies a two-parameter model to
describe the mechanical properties of cortical bone. Equations
(10)-(12) reveals that the Young’s modulus, ultimate stress and
toughness are influenced by these two variable parameters α
and γ . Figure 4 shows a 3D curve surface of these three
equations. As indicated in Fig. 4(a), the value of E rises with
an increasing of bone volume fraction α or bone mineral
fraction γ . The Young’s modulus is thus positively related to
α and γ . Figures 4(b) and (c) show that ultimate stress and
toughness are also positively related to volume fraction α,
while inversely related to bone mineral fraction γ. Table 2 lists
the levels of influence exerted on Young’s modulus, ultimate
stress and toughness by α and γ . The Young’s modulus, E,
changes in the same direction as changes in α and γ . When
α increases by 5%, E rises by 13.25%. If α decreases by 5%,
then E falls by 12.26%. Conversely, when γ increases by 5%,
E only rises by 1.62%. When γ decreases by 5%, E falls by
1.68%. The relationship of the bone volume fraction with
Young’s modulus is more significant and important than its
relationship with the mineral fraction γ.
As regards ultimate stress, when α increases by 5%,
ultimate stress rises significantly by 19.78%. However, when
γ increases by 5%, ultimate stress falls by 2.41%. Thus α
has a stronger effect on ultimate stress than γ , and α and
ultimate stress are positively related. Conversely, γ shows a
small inverse effect on ultimate stress. Toughness is found by
calculating the total area under the stress-strain curve from
zero stress up to maximum stress. When α increases by 5%,
toughness rises by 9.98%. However, when γ increases by
5%, toughness falls by 3.26%. Thus α also has a stronger
effect on toughness than γ , and α and toughness are also
positively related. When both bony matrix volume and mineral
content rise, in other words, when bone mineral density
increases, the mechanical properties of the whole unit still tend
to strengthen.
Mechanical Properties of Bone
Table 3:
5
Comparison of literature regarding relations between Young’s modulus and volume fraction ( α ) and ash fraction ( γ ).
Source
Loading condition
Material
compression
Currey [22]
Schaffler and Burr [12]*
Hernandez et al. [24]
Current study
Young’s modulus exponent
volume fraction ( α )
mineral fraction ( γ )
cortical and cancellous
2.58±0.022
2.74±0.129
tension
cortical
3.52
3.17
tension
cortical
10.92
3.91
compression
cortical
2.55
0.33
* Equation based on individual one-parameter model.
As stated above, the solid skeleton in bone tissue consists
of an organic and inorganic phase. The inorganic component of
bone makes the tissue hard and rigid, whereas organic
components account for the flexibility and resilience of the
bone [32]. That is, if the total weight of a solid skeleton is
fixed, increasing the mineral fraction of bone decreases the
flexibility of bone tissue, strain and elongation. Toughness, the
area under the stress-strain curve, is thus decreased. Several
studies have only investigated the relationship between
Young’s modulus and bone components [12, 22, 24].
Documents concerning the quantitative effects of bone mineral
fraction and bone volume fraction on toughness or ultimate
stress have not been found. In this study, the measured bone
volume fraction, bone mineral fraction, and Young’s modulus
are all within the ranges found in the literature [24, 32]. Thus,
the obtained toughness and ultimate stresses in this study are
reliable. Generally, increasing bone mineral content improves
the mechanical properties of bone; however, that may be due
to an increase in the mineral content, the total volume of solid
skeleton increases in the meantime, and thus the porosity of
the cortical bone decreases. This study showed that bone
volume fraction has a stronger positive effect on mechanical
properties than bone mineral fraction. This analytical result is
also consistent with those obtained by previous studies [12, 22].
Table 2 indicates that when bone volume fraction is fixed, the
changes of the value for E, σ ult and the toughness as the
value for bone mineral fraction fluctuates by itself. Notably,
the specimens used in this study were harvested from 2– to
3–year-old bovine femurs. The analytical results of this study
cannot be extended to newborns or premature subjects.
Schaffler and Burr [12] drew the same conclusions from
their experiment, indicating that mineralization directly
influences cortical bone stiffness, and Young’s modulus of
cortical bone is more highly related to volume fraction. Power
law models based on bone volume fraction and mineral
fraction have also been applied in previous studies (Table 3).
Hernandez et al. [24] used specimens from human bones for
the compression tests. The volume fraction in their study was
obtained from a derived expression for bone volume fraction
as a function of tissue density, rather than from direct
measurements. The specimens adopted in his experiments
included cortical bones and cancellous bones, with volume
fraction ranging from 2.2% to 84.3%. While Currey [22]
performed his tension tests with cortical bone specimens
obtained from different animal species. The γ parameter in
his study denoted the calcium content, not the mineral fraction.
Schaffler and Burr [12] obtained their bone specimens
from 2–3-year-old cattle, with volume fractions ranging from
92.2% to 97.1%, clearly well above the normal value found in
other bovine bones normally applied for experiments.
Furthermore, because Young’s modulus was discussed
separately from volume fraction and mineral fraction, a larger
exponent related to bone volume fraction might have been
reported. The cortical bone specimens applied for compression
tests in this study were obtained from 2–3-year-old adult
bovine, with a volume fraction ranging from 84.6% to 88.2%.
Since different specimens resulted in a different volume
fraction range, the obtained exponent values were not
consistent. Schaffler and Burr [12] found that cortical bone and
cancellous bone are made of the same materials, but with
different structures and significant differences in volume
fractions, and different mechanical properties, and hence
cannot be properly described by a single mathematical model.
Hernandez et al.’s experimental results [24] indicate that the
mineral fraction has a stronger effect than the volume fraction
on Young’s modulus. This finding is different from those from
other studies, possibly because Hernandez et al. adopted
specimens including both bones and cancellous bones with a
wide volume fraction range. A review of the experiments
results in Table 3, and excluding Hernandez’s findings,
indicates that all other data show some differences, but all
consistently reveal that the volume fraction exponent is
typically larger than the mineral fraction exponent, suggesting
that the relationship between Young’s modulus and the volume
fraction is more significant than the mineral fraction.
Day et al. [33] found that when bisphosphonates were
applied to treat osteoporosis in dogs, the bone mineral density
rose significantly increased during the first year of treatment,
and rose more slowly thereafter. The initial BMD gain is due
to the increase in total bone volume (when the bone
remodeling space was filled in), while the later BMD gain is a
result of increased mineralization. Day et al. [33] also
observed that the increase in Young’s modulus was caused by
increased bone mass and altered trabecular architecture, rather
than changes in the calcified matrix. This finding is consistent
with the results of this study, suggesting that the Young’s
modulus correlates more strongly with the volume fraction
than with the mineral fraction.
This study has a few limitations. First, human bone
specimens are hard to obtain, so bovine cortical bones were
J. Med. Biol. Eng., Vol. 26. No. 1 2006
6
adopted instead. Should an opportunity occur in the future, we
shall consider using human bones for the experiments in order
to obtain more comprehensive data. Second, only compression
tests were performed. If tension and bending tests could be
carried out on specimens from different parts of the bone with
different orientations, then more human bodily movements
could be simulated in various positions, thus obtaining more
practical and precise data than could be obtained in this study.
Third, the specimens used in this experiment were all obtained
from healthy normal cattle, with similar bone structures and
mechanical properties. Future work may investigate bone
resorption phenomena and identify changes to the mechanical
properties that occur in response to some disease states, such
as osteoporosis and abnormal hormone levels, with the aim of
developing diet programs or medicines that could help patients
restore their BMD and increase the mechanical strength of
bones.
Conclusion
The percentage of the solid materials content in cortical
bone has a much more significant effect than mineral content
on the mechanical properties of bone, such as Young’s
modulus, ultimate stress and toughness. This investigation
further found that volume fraction is positively related to all
these three mechanical properties, and it is the most important
factor among those studied. Conversely, the mineral fraction is
inversely related to ultimate stress and toughness, and has a
smaller effect than the volume fraction. When both bony solid
matrix volume and mineral content rise, that is, when the bone
density increases, the mechanical properties of the whole unit
still tend to strengthen. The cortical bone mechanical
properties, Young’s modulus, ultimate stress, and toughness,
which are required parameters for biomechanical calculations,
could easily be predicted from the equations derived in this
study, together with the volume and mineral fractions.
Acknowledgement
The authors would like to thank Chung Shan Medical
University for financially supporting this research under
Contract No. CSMU92-OM-B-036.
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
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