The quite scattered experimental results revealed the inhomogeneous mechanical properties of cancellous bone in human cervical vertebral bodies. This phenomenon was also observed experimentally for vertebral bodies from lumbar spine [17–19]. Fresh cancellous bone can be considered as a two-compartment tissue composed of mineralized hard tissue and
Static compression tests on cancellous bone
0 2 4 6 8 10 12 14 16 18
Compressive Strain (%)
Compressive Stress (MPa)
Static tensile tests on cancellous bone
0 1 2 3 4 5 6 7 8 9
0 10 20 30 40 50 60
0 5 10 15 20 25 30
Tensile Strain (%)
Tensile Stress (MPa)
a
b
Fig. 16.2 The complete stress-strain curves of cancellous bone under (a) compression and (b) tension loading
112 J.F. Liu et al.
inter-trabecular marrow. Nafei et al.[20] studied the architectural properties of trabecular bone, and concluded that bone volume fraction was the major predictor of its compressive mechanical properties. The contribution of marrow to specimen’s overall resistance comes from its relative motion ability among pores during testing. Only at the strain- rate beyond 10/s, the presence of marrow increased the compression strength, modulus, and energy absorption of cancellous bone [21]. In the current study on cancellous bone with open cell structure and at strain rate of only 103/s, the marrow has sufficient time to flow among cells freely; therefore the mechanical resistance of fresh bone was only attributed to the bony structure of mineralized hard tissue, which could be related to the factor of apparent density alternatively (rA).
Apparent density is defined as division of hydrated bone tissue mass by specimen volume calculated from its initial dimensions. Measuring the apparent density of fresh bone need to remove the inside marrow completely. In this study, the fresh bone specimens were defatted with a ultrasound bath of trichloroethylene solvent effectively; then re-hydrate it in saline solution and measure its submerged weight; the hydrated bone tissue mass was obtained by weighing it in air after centrifuging the specimen for removing the inside water; the bone tissue volume can be calculated from the submerged and hydrated bone tissue mass in line with Archimedes’ principle. From the tested compression specimens in this study, the derived mean value (standard deviation) of apparent density was 0.5589 (0.2284) g/cc; bone tissue density was
0 2 4 6 8 10 12 14
Compressive Strain (%)
Compressive Stress (MPa)
ρF=1.2785g/cc;
ρA=0.9247g/cc
ρF=1.1117g/cc;
ρA=0.5765g/cc
ρF=0.5988g/cc;
ρA=0.2964g/cc
0 1 2 3 4 5 6 7
0 0.5 1 1.5 2 2.5 3 3.5
0 2 4 6 8 10
Tensile Strain (%)
Tensile Stress (MPa)
ρF=1.2478g/cc;
ρA=0.6516g/cc
ρF=0.8964g/cc;
ρA=0.3909g/cc
ρF=0.5558g/cc;
ρA=0.2383g/cc a
b
Fig. 16.3 Illustration of (a) compression and (b) tension stress-strain curves before failure for specimens of low, middle and high density. (solid line: experimental results;dot line: polynomial fitting;line with markers: redrawing by nonlinear models based on the same value ofrFandrA) 16 Quasi-static Compressive and Tensile Tests on Cancellous Bone in Human Cervical Spine 113
2.1192 (0.2796) g/cm3, which is the division of hydrated bone tissue mass by bone tissue volume; and bone volume fraction was 27.7140% (9.8033%), which is the ratio of bone tissue volume to specimen’s total volume.
Although apparent density is more reasonable in characterizing bone mechanical properties, it is regretful that it can only be measured after testing for each individual specimen because the chemical solvent may alter bone’s properties. This leads the impossibility to obtain the apparent density for tensile specimen, whose ends are normally embedded in adapters with bone cement or adhesives. Therefore in this study, the fresh bone density (rF), which was defined as the division of wet specimen’s mass by its total volume, was measured before testing for all compression and tension specimens. The wet specimen’s mass was measured by a precision digital balance after the specimen’s surfaces were blotted dry on absorbent paper for few seconds; and the total volume was calculated from its dimensions measured by digital caliper.
For the reason of visible big-sized voids on the surface of cancellous bone, the specimen’s total volume is not measured by the Archimedes’ principle, from which the derived density is named as “actual” density [22]. Fresh bone density has the advantage that it can be measured before testing for all specimens without affecting their tissue’s properties. The derived mean value (standard deviation) of fresh bone density in this study is 1.0680 (0.2534) g/cc. An exponential relationship seems exist between measured apparent density and fresh bone density,
rA¼0:1062e1:4538rF (16.3)
with an additional virtual point by assuming these two densities reach identical at the value of 2.1192 g/cm3(the mean value of bone tissue density). The fitted exponential curve seems able to present the tendency quite well as shown in Fig.16.5.
Therefore in an inverse way, the apparent density of tensile specimen could be derived from this equation approximately, and the mean value is found to be 0.4174 (0.1449) g/cc in this study.
Elastic modulus, ultimate stress and ultimate strain are the three main parameters for reflecting the mechanical properties of porous material, and can be related to density. Previous researches on cancellous bone from various locations of human or animal showed, that these parameters are usually regressed with density by empirical equations, such as linear equation (Y¼a + br) [11,23], power equation (Y¼arb) [10,24–27], or both relationships [28–30]. Assuming slabs of cancellous bone resemble cortical bone more closely, Carter, et al. [21] linked cancellous bone properties with that of cortical bone using the functionss/sc¼K(r/rc)2and E/Ec¼K(r/rc)2. In this study, the whole experimental values of these parameters are plotted against the densityrFandrArespectively and fitted by power and linear equation for comparison. As shown in Fig.16.6, the compressional modulus or ultimate stress follows the power relationship withrFclosely, and are poorly fitted by linear equation; but the tendency changes to more linearity and can be fitted well by both relationships if plotted against rA. The tensile modulus or ultimate stress can be fitted well with the power or linear equation whenever plotted againstrFor
0 1 2 3 4 5 6 7
0 1 2 3 4 5 6
Strain (%)
Stress (MPa) Tension
Compression
Mean (SD) ρF (g/cc) E (Mpa) εu (%) σu (MPa) εy (%) σy (MPa) Compression 1.07(0.25) 286.57(156.72) 3.13(0.83) 6.78(3.65) 2.23 5.71
Tension 0.88(0.26) 107.86(30.58) 6.76(2.62) 4.24(1.89) 2.85 2.86
Fig. 16.4 Represent of overall average stress-strain curve and the mean properties of cancellous bone in tension and compression
114 J.F. Liu et al.
rA (note that the apparent density for tension specimens is not experimental value, but derived value from Eq.16.3).
Although the fitting quality seems well, linear relationship is more empirical and may be affected by the adopted density range. Power relation has its mechanical basis; by extending the knowledge from engineering cellular materials, Gibson [31]
theoretically predicted the deformation mechanisms of several cell models for different kind of cancellous bone, and drew out the corresponding power laws (with index between 1 and 3) which agreed well with experiment data cited; Kasra and Grynpas [32] predicted the power equation (E¼1737r1.78) between the modulus (MPa) and apparent density (g/cm3) for cancellous bone in human vertebral body by finite element analyses with a cubic model of rod-like cell structure.
For clarity, Table 16.1 lists all the fitting equations for modulus and ultimate stress against densities. From these equations, empirical constitutive equations could be derived to describe stress varying against strain and density. The linearity constitutive equation
s¼Ee¼EðrÞe (16.4)
is the simplest style and could be obtained by directly substituting E(r) from that listed in Table16.1for compression and tension. The linearity models may be able to describe the compression stress-strain curves well for its more linearity than tension curves. In this study, more emphasis will be put on modelling the nonlinear profile of stress-strain curves. From Fig.16.6, it can be seen that the ultimate strain almost does not correlate with density whether it is plotted againstrForrA; and the compression failure strain seems more constant than tensile value for its less scattering. This independent characteristic of ultimate strain on density has also be observed by more other researches, such as Keaveny et al. [11] in testing of bovine proximal trabecular bone and Giesen et al. [25] in testing of cancellous bone in human mandibular condyle.
Because the profiles of stress-strain curves under different density in this study are similar despite of their magnitudes, this independent feature of ultimate strain could help to make a hypothesis that the magnitude ratio of stress-strain curves at every stain value before failure is approximately equal to the ratio of their ultimate stressessu. If using the mean stress-strain curve (fm(e), as indicated in Eq. 16.1) as the reference curve, then the empirical nonlinear constitutive equation can be expressed as :
s¼ðr r= mÞafmðeÞ (16.5)
where the power relationship between ultimate stress and density is adopted;ais the power index as listed in Table16.1, and rmis the mean density in this study corresponding tofm(e)in compressive or tensile tests. Substituted the known items in, the nonlinear constitutive equations could be obtained for compression and tension against fresh bone density or apparent density, as expressed in Table16.1. An illustration of these models to predict the stress-strain curves under compression or tension is shown in Fig.16.3, by using the same density value of the plotted experimental results of specimens in low, middle and high density. It is seen that these calculated curves could predict the nonlinear trend of compression or tension stress- strain curve quite well till the strain limit, whenever in term of fresh bone density (s¼f(rF,e), plotted in line with square marker) or apparent bone density (s¼f(rA,e), plotted in line with “” marker). The closer of the experimental ultimate stress to the fitted curve in Fig.16.6, the closer of the experimental curves to the predicted curves. It should be noted that the experimental results are scattered, caused by various factors such as cadaver individual particulars, bizarre pores in specimen etc. or even experimental errors. Therefore, the predicted curves, which only take the main factor of density into
0.0 0.5 1.0 1.5 2.0 2.5
0.0 0.5 1.0 1.5 2.0 2.5
Fresh bone density (g/cc)
Apparent density (g/cc)
Fig. 16.5 The exponential relationship of apparent density with fresh bone density for cancellous bone from human cervical spine
16 Quasi-static Compressive and Tensile Tests on Cancellous Bone in Human Cervical Spine 115
consideration, are no need and impossible to coincide with all the experiment curves well. These empirical constitutive equations give a good prediction of stress-strain curve along the mean variation tendency against density.