Linear correlations of the elastic coefficients, engineering moduli, anisotropy ratios and axial shear damping ratio with the apparent mass density were investigated.
The results (correlation coefficient, significance, slope, intercept) are reported in Table 5.1. All the parameters are significantly correlated to the apparent density (allp <0.05). The elastic coefficients and engineering moduli increase with density, while anisotropy and Poisson’s ratio ν31 andν13 decrease with density.
To investigate the influence of the age (69−94 years) and sex of the donors, the medians of the elastic parameters and mass density were calculated for each donor over the two to four specimens corresponding to different locations. This reduces the data set to one value of each parameter per donor. These median values showed no linear correlation (Pearson’s) with age (allp >0.25) nor rank correlation (Spearman’s; all p > 0.1). Differences between sexes were investigated using a non-parametric Wilcoxon rank-sum test. No differences were found (all p >0.05).
A Kruskal-Wallis test (non-parametric equivalent to classical one-way analysis of variance) was done for each elastic parameter and for the density, with the specimens grouped according to their anatomical position. A significative difference (at a 0.05
5.4. Statistical analysis 105
Figure 5.11: Results of the three parameters anisotropic viscoelasticity analysis for seven selected specimens. Despite some large uncertainties, anisotropy is clear. The damping is stronger (Q−1is large) for the transverse parametersE1 andC66.
1 2 3 4 5 6 7 8 9
Figure 5.12: Predicted quality factors of the peaks, after fitting the viscoelastic parameters, compared to the measurents, for two specimens. Error bars on the mea-surement assume 5%uncertainty. The variations of the peak widths are well captured by the model, whether the Poisson’s ratios are included in the analysis or not.
confidence level) in the median value was found for only one parameter: the stiffness coefficientC13. That stiffness coefficient was larger for the position C-1/3 than the for the position C+2/3.
RUS technique and hierarchical modeling
In the previous chapters, the applicability of RUS to highly attenuative materials was established for a single cortical bone specimen (Bernard et al.,2013, Chapter2) and a few specimens of plastic materials with damping and stiffness comparable to that of bone (Bernard et al., 2014, Chapter 2). The present study is the first one to report the measurement of bone anisotropic elasticity using RUS on a large collection of specimens. Moreover, to the best knowledge of the author, this work involves the largest specimen collection ever characterized using RUS in a single study, whatever the research context (material sciences, geophysics, biomechanics).
The application on a large collection was made easier by the automated reso-lution of the inverse problem, based on a Bayesian framework. Resoreso-lution of the inverse problem usually requires a tedious manual pairing of the resonant frequencies by the user, or the use of a complex experimental setup to measure the shapes of the resonant modes in addition to their frequencies (Ogi et al., 2002). Both method-ologies limit the number of specimens measurable in a given amount of time. In contrast, the method described here does not require the user to play a role during the inversion, nor to modify the original simple RUS setup.
The Bayesian inversion scheme developed in Chapter 4 requires a normal prior distribution on the elastic parameters. Here, the approach was extended to a col-lection of specimens through hierarchical modeling. All the specimen are processed together, and the distribution of parameters in the collection is used as a prior for each specimen. To infer the properties of the population distribution, a nonifor-mative prior has been used for the means, and a weekly infornonifor-mative prior for the covariance matrix, so that the final estimates are not biased by information indepen-dent from the measurements. The final estimates appears to be weakly sensitive to a change in the prior distribution of the covariance matrix. Indeed, both changing the number of degree of freedom ν0, and scaling up the matrix Λ0 by a factor 2 or 3, did not affect the results significantly (i.e. produced a deviation larger than the confidence interval).
The hierarchical approach proves to be necessary here, because the available experimental elastic parameters from which a prior distribution could have been deduced were different from the parameters of the specimen collection at hand.
Indeed, the mean values of the elastic coefficients inGranke et al.(2011) wereC11=
5.5. Discussion 107
19.5,C33= 29.2,C44= 5.7, and C66= 4.2 GPa, while there are C11= 14.0,C33= 26.2,C44= 5.3, andC66= 3.4GPa for the present population. Using the empirical distribution would than have biased the results towards higher stiffness, particularly for the less stiff specimens. A simpler alternative to hierarchical modeling could have been to process first a few representative specimens with manual pairing of the frequencies (Chapter2) to deduce a prior distribution, process all the specimens with that prior, and then possibly update the prior and re-process the whole population.
Such a methodology is known as empirical Bayes, and is only an approximation of the full hierarchical approach.
The hierarchical modeling suppose that all the specimens are samples of a single distribution, and that this distribution is normal. Both hypothesis are arguable.
First, the specimens were harvested on close but different locations, on donors of both sexes and with different ages. None of this parameters was found to cause significative differences on stiffness in that study (except for only one stiffness coef-ficient: C13). The assumption of a single distribution is then justified by the results.
This might not be the same in, for example, studies comparing two or more popula-tions (e.g. healthy/unhealthy). Secondly, the normal hypothesis might not be fully adapted. Indeed, the empirical distributions of the elastic parameters were found to deviate from normal distributions, due the presence of a few soft specimens whose mass densities were lower than 1750 g/cm3 (Fig. 5.8). Hence it might have been more adapted to drop the normal hypothesis, and the associated mathematical con-venience, to use an alternative distribution (e.g. a long-tailed distribution such as the Student’s distribution). This was not pursued in this work.
The precision of the results, quantified by the half-width of the 68%confidence intervals deduced from the posterior probability distribution, is 1-2% for the shear coefficients and 2-3%for the Young’s moduli (median values). The other coefficients are determined with a lower precision: 4% for C11, 7% for C33, 11% for C13 and 8-10% for the Poisson’s ratios. This ordering of the precision, depending on the nature of the elastic parameter, is consistent with results reported in the previous chapters and the RUS literature in general (e.g. Migliori and Sarrao, 1997; Landa et al., 2009). The precision is however slightly worst in the present study than in Chapters 2 and 3. This is in part due to the inversion method, which does not assume the pairing to be known here. Therefore, the posterior distribution of the elastic parameters is a compound of several distributions for several probable pairings, which gives more or less complicated posterior distribution depending on the specimen.
Another reason of the degraded precision is the slightly worst agreement between predicted and measured frequencies, reflected in the median value of the maximum a posteriori estimates of σ (0.83%), which is larger than the RMS errors reported
in Chapter 2 for a bone specimen (0.3%) or in Chapter 3 for two bone mimetic material specimens (0.42 and 0.45 %). This has several potential causes:
• The geometry of the specimens might differ from a perfect rectangular paral-lelepiped. Several specimens were harvested from the same bone cross-section, which required several repositioning of the section during cutting. Although alignment was carefully checked, some geometrical imperfection inevitably oc-curred. Deviations from perpendicularity of about 1◦ or larger can shift the frequencies of a few tenth of percents (Migliori and Sarrao,1997).
• Also because of cutting, the bone axial direction might be tilted with respect to the edges of the specimens. A tilt angle of a few degrees can shift the frequencies of a few tenth of percents (Migliori and Sarrao,1997).
• Due to the large quantity of data, the signal processing – which is still a user-dependent step – might not have been done as carefully than is was previously done for a few specimens.
• The transversely isotropic symmetry might not be an accurate model for all the specimens. Although Espinoza Orias et al. (2008) and Granke et al. (2011) found weak or nonexistent statistical differences in elastic properties in the plan orthogonal to the bone axis for femur mid-diaphysis, some anisotropy might still occur for individual specimens, for example due to a gradient in density between the inner an outer parts of the cortical shell.
• The forward model assumes homogeneity of the material, while bone is fun-damentally inhomogeneous. RUS has been shown to be weakly sensitive to small linear gradients of elastic properties (Seiner et al.,2012), but this might not be an accurate model for bone. Ulrich et al. (2002) studied the effect of a localized mass perturbation (that could be a large pore in bone), and found that a frequency shift up to 0.2%could be observed.
All the arguments above might also explain why no satisfactory agreement between predicted and measured frequencies could be obtained for four of the specimens.
Viscoelasticity entails a variation of the real and imaginary part of the elastic parameters with frequency (Lakes,2009). Hence, because the frequency of the first peak spans a large range (100-400 kHz), it was questionable whether the width of the first resonant peak was dependent on its central frequency. No linear correlation was found between these two quantities (p=0.99), which indicates that the effect of frequency on the viscoelastic parameter is smaller than the effects of bone structure and composition. Moreover, Qf irst was not correlated to the relative sensitivity of the first peak to the axial shear stiffnessC44(p=0.64). Therefore, the approximation made to estimate the axial shear viscoelaticity from the first peak alone did not
5.5. Discussion 109
Table 5.1: Linear correlation of the elastic coefficients and engineering moduli (GPa, except Poisson’s ratios), viscoelastic parameter Q44 (no dimension) and anisotropy ratios (no dimension) to the apparent mass density (g/cm−3).
Parameter Corr. coef. r2 p-value Slope Intercept Range C11 0.80 <10−5 28.9 -39.4 5.3 - 20.3 GPa C33 0.85 <10−5 30.6 -30.8 16.8 - 31.1 GPa
C13 0.53 <10−5 7.0 -4.8 5.1 - 11.0 GPa
C44 0.89 <10−5 9.4 -12.2 2.4 - 7.2 GPa
C66 0.87 <10−5 8.0 -11.3 1.2 - 5.0 GPa
E1 0.86 <10−5 22.7 -32.3 3.1 - 14.3 GPa E3 0.80 <10−5 33.5 -42.7 7.9 - 25.4 GPa
ν12 0.11 <0.05 0.13 0.14 0.21 - 0.49
ν13 0.25 <10−4 -0.11 0.40 0.14 - 0.30 ν31 0.72 <10−5 -0.61 1.54 0.29 - 0.68 C33/C11 0.60 <10−5 -2.5 6.58 1.35 - 3.15 C44/C66 0.77 <10−5 -1.3 4.01 1.38 - 2.17 E3/E1 0.54 <10−5 -2.0 5.85 1.67 - 3.31
Q44 0.69 <10−5 31.8 -31.6 17.3 - 31.1
introduce a bias. This is a very straightforward analysis, not even requiring to estimate the elasticity.
The anisotropic viscoelastic analysis revealed a significant anisotropy in both shear and Young’s moduli. The agreement between measured and predicted peak widths was about 5%, which is comparable to the precision of the measurement, and comparable to previous reported results for similar analysis on low damping materials (Seiner et al.,2013). We observed that the contribution of the imaginary part of the Poisson’s ratios was negligible given that precision, which means that a simple three parameter model can be used, requiring only a few measured peak widths to lead reliable results.
Tibial cortical bone viscoelasticity
All the elastic coefficients, engineering moduli, and anisotropy ratios show a sta-tistically significant dependence to the apparent mass density (Table 5.1, Fig. 5.8 and Fig. 5.9). This dependence is strong (r2 > 0.5), except for ν12 (r2 = 0.11) and ν13 (r2 = 0.25) which are almost constant functions of density. The elastic coefficients Cij, the Young’s moduli and the Poisson’s ratio ν12 increase with mass density, while the two other Poisson’s ratios and the three anisotropy ratios decrease with mass density. Such a strong dependence of bone stiffness to the apparent mass
density was expected. Mass density is indeed known to be a good predictor of bone millimeter-scale stiffness, both for trabecular (Carter and Hayes, 1977) and cortical tissues (Schaffler and Burr, 1988;Keller,1994). Linear or power law rela-tions between density and stiffness are used in studies of bone mechanics at organ scale to set the stiffness in the numerical models from density measurement (Schileo et al., 2007, 2008;Austman et al., 2008). Measuring femoral bone elasticity using ultrasound waves, Espinoza Orias et al. (2008) also observed correlations between apparent mass density and longitudinal stiffness, with similar strength, but no corre-lation with shear stiffness. Keller(1994) found a strong linear correlation (r2= 0.69) between the Young’s modulus (mechanical testing) and dry mass density in human femoral bone.
The observed decrease in longitudinal stiffness anisotropy ratioC33/C11with in-creasing density is also consistent with the findings ofEspinoza Orias et al. (2008), who reported a slope of -2.95 (r2 = 0.61), close to the present result (Table 5.1).
Moreover, mechanical models relating bone elasticity to porosity and micro-scale elastic properties predict an increase of the anisotropy with increasing porosity (Parnell and Grimal,2009;Parnell et al.,2011). As density and porosity are in first approximation inversely related, the observed variation in anisotropy is consistent with the models. However, the longitudinal stiffness anisotropy ratios C33/C11 re-ported here are larger than previously measured or predicted values for femoral bone: Espinoza Orias et al. (2008) found C33/C11 in the range 1.25 - 1.75, and Parnell et al. (2011) predicted 1.3 - 1.5 for the physiological range of porosity. We observe a larger range of 1.35 - 3.15. This might be due to physiological differences between tibia and femur, but could also be partially explained by the large range of mass densities in our specimen collection. To our knowledge, experimental observa-tions of the decrease in the two other anisotropy ratios – C44/C66and E3/E1 – are not reported in the literature. Overall, the increase in anisotropy with decreasing density indicates that the transverse elastic properties (C11,E1, andC66) are more sensitive than the axial properties (C33,E3, and C44) to bone material losses. This might be relevant for bone quality assessment using quantitative ultrasound meth-ods. Physiologically, it is consistent with the function of the tibia, which withstand mainly axial stresses.
Two of the Poisson’s ratios – ν12 and ν13 – were found to vary weakly with apparent density (Table 5.1 and Fig. 5.8). This is consistent with the model of Parnell et al.(2011), which predicts small variations in Poisson’s ratios as a function of porosity. The mean values are ν12 = 0.397 and ν13 = 0.191, in good agreement with values previously measured on femur and tibia using ultrasound wave velocities, reported in Table 5.2. The third Poisson’s ratio ν31 is related toν13 trough
ν31 E3 = ν13
5.5. Discussion 111
Table 5.2: Mean Poisson’s ratios measured in this study compared to values of the literature, measured using ultrasound wave velocities. Standard deviations are indi-cated in parenthesis, if available. In all studies, direction 3 is the axial direction of bone. Our study assumes a transversely isotropic symmetry, which impose equality of pairs of Poisson’s ratios.
This study Ashman et al. (1984) Rho (1996)
Bone Tibia Femur Tibia
Number of specimens 55 60 96
ν12 ν12=ν21 0.376 0.420 (0.075)
ν21 =0.397 (0.042) 0.422 0.435 (0.057)
ν13 ν13=ν23 0.222 0.237 (0.041)
ν23 =0.191 (0.024) 0.235 0.231 (0.035)
ν31 ν31=ν32 0.371 0.417 (0.048)
ν32 =0.390 (0.077) 0.350 0.390 (0.021)
Therefore, the observed variations of ν31 with density reflects the variation of the anisotropy ratio E3/E1 (Fig. 5.8).
Bone viscoelasticity is difficult to characterize using ultrasound methods based on wave propagation, because these methods can not usually separate two cumula-tive effects of energy losses: intrinsic viscoelastic losses and scattering effects (Lakes et al.,1986;Haïat,2011). In contrast, RUS is free from any scattering effect, and is only sensitive to the intrinsic viscoelastic losses (neglecting the energy leakage due to contact with air and with the transducers, as discussed in Chapter3). This ad-vantage makes RUS a valuable tool to study the phenomena responsible for energy dissipation in bone. To the best of our knowledge, the dependence of bone axial shear damping to mass density was not reported before. However, a negative corre-lation of broadband ultrasonic attenuation with bone apparent density and mineral density was observed in bovine cortical bone (Sasso et al.,2008), which is consistent with our result.
Anisotropic viscoelasticity for the Young’s modulus has been previously observed in bovine cortical bone using mechanical tests (Iyo et al., 2004). Here, anisotropy is observed in both shear and Young’s viscoelastic moduli. It is not clear which physical phenomena are responsible for the viscoelastic behavior of bone (Lakes and Katz,1979), particularly at ultrasonic frequency. Among several potential causes we can mention fluid motions inside the pores, thermomechanical effects and motions at interfaces (e.g. motion of at the cement lines between osteons or motion of the collagen lamellae inside osteons). Due to the structural anisotropy of cortical bone, all these phenomena are expected to entail an anisotropic viscoelastic behavior.
No effects of age or sex were found on bone apparent density, elasticity, anisotropy and damping. However, the number of donors (6 males, 13 females) might be too low for any effect of the sex to be observable, and the age range (69 -94 years) was quite narrow. The influence of these two parameters is worth being investigated further using RUS. As a mechanism of energy dissipation, viscoelatic-ity is intimately related to crack initiation and propagation in bone (Schapery, 1975; Ritchie et al., 2009), and bone strength and toughness have been found to be rate-dependent (Carter and Hayes, 1977; Currey, 1988a). Crack-initiation and crack-growth toughness in human cortical bone are also age-dependent (Nalla et al., 2006;Zimmermann et al.,2011). Using nanoindentation, Wu et al.(2012) observed a difference in creep time in cortical bone for human females and males.
RUS was for the first time applied on a large collection of cortical bone specimens.
The full stiffness tensor was obtained under a transversely isotropic symmetry as-sumption for 55 out the 59 specimens, with a precision ranging from about 1%(axial shear stiffness) to about 10%(diagonal stiffness coefficient). This demonstrate that RUS is a valuable tool for a routine assessment of bone anisotropic elasticity on millimeter-sized ex vivo specimens. Additionally, the shear viscoelasticity in the axial direction of bone was characterized for all specimens from the first resonant peak, and three viscoelastic parameters could be obtained for some specimens.
We observed that bone elasticity is strongly dependent on bone apparent mass density, which was expected. The axial shear viscoelastic parameter also correlates strongly with mass density. Anisotropy of the viscoelastic behavior was observed:
the Qratios of the real to the imaginary part of the moduli are larger in the axial direction of bone, both for Young’s and shear moduli. This might help to better understand the phenomena responsible of bone viscoelasticity, which is intimately related to bone fracture mechanisms.
In the view of the promising results on the anisotropic damping, it might be worthwhile to reformulate the estimation of the damping parameters in Bayesian terms. The introduction of prior information might allow a reliable inversion for more than a few selected specimens, and then permit to investigate the variation of damping anisotropy with bone apparent density. About the precision of the elastic coefficients estimation, the main limiting factors are the imperfect geometry of specimens and the limited number of measured frequencies. The first difficulty could be overcome by a more complex cutting protocol, or by measuring and tacking into account the imperfections of the specimens in the forward modeling (e.g. using finite element modeling). The second difficulty is intimately related to the strong damping of ultrasound in bone. Better signal processing methods could perhaps
5.6. Conclusion 113
be developed to find more resonant frequencies. Particularly, the spectrum were processed independently of any information on the material properties and specimen shape, while it is clear that this information could help the estimation of the peak parameters.
Summary and conclusion
Deep understanding of the structure-function relationships of cortical bone in the context of bone quality assessment is still missing. Cortical bone millimeter-scale elasticity is an important aspect of bone quality but is difficult to measure, even ex vivo. Currently available methods – mechanical testing and ultrasound waves velocity measurements – have limitations arising mainly from the anisotropy of the tissue, its heterogeneity, and the small size of the specimens. In this work, reso-nant ultrasound spectroscopy (RUS), a method particularly suitable for anisotropic elasticity assessment on small specimens, has been adapted to highly attenuative materials such as cortical bone. Additionally, RUS also provided insight into the viscoelastic behavior of bone.
RUS estimates the elastic properties from the free vibration resonant frequencies of a single millimeter-sized specimen, and the viscoelastic damping from the width of
RUS estimates the elastic properties from the free vibration resonant frequencies of a single millimeter-sized specimen, and the viscoelastic damping from the width of