5.2 Materials and methods
5.2.4 Viscoelasticity
Viscoelasticity was modeled by adding an imaginary part to the stiffness tensor, as in Chapter 3:
Cij∗ =Cij +iCij0 =Cij(1 +iQ−1ij ), (5.22) whereQ−1ij are the ratios of the imaginary to the real part of the stiffness constants.
If the inverse damping ratios Q−1ij are small (Q−1ij 1) they can be linearly related
5.2. Materials and methods 99
to the inverse of the peaks quality factorsQ−1k (Leisure et al.,2004;Ogi et al.,2003):
Q−1k =X
Where the coefficients of the linear system are the normalized relative sensitivities of the resonant modekto the elastic coefficients, i.e.
X
When the elastic constants Cij have been estimated, Eq. (5.23) can be inverted to yield the damping ratios Q−1ij from the measured quality factors Q−1k , as done in Chapter3.
However, this analysis presents two difficulties: 1) the peak quality factors Qk
are difficult to estimate precisely particularly when overlapping is strong. More-over, as the relative sensitivities of some frequencies to some elastic modulus are low, the inversion is badly conditioned and sensitive to errors in the measuredQk. 2) Eq. (5.23) is dependent the pairing of the frequencies, which in our approach is not fixed. A good approach would then be to solve Eq. (5.23) for each pairing and set of elastic parameters sampled during the exploration of the posterior prob-ability distribution, which would lead to a distribution of viscoelastic parameters.
Perhaps an even better solution would be to formulate the inversion of Eq. (5.23) in a Bayesian framework, with the introduction of a prior distribution for theQ−1ij . None of these two approaches was pursued in these work, but we went for simplified analysis based on the selection of the most reliable measurements of the quality factorsQk.
Analysis based on the first resonant mode
The first vibration mode of rectangular specimen with small aspect ratios is usually a pure shear mode (Migliori and Sarrao,1997), because shear waves have a smaller velocity than longitudinal waves. Moreover, many of the specimens in that study are longer in the third direction (axial direction of bone), so that the first resonant mode depends mainly on the axial shear modulusC44. Hence, the mechanical quality factor of the first resonant modeQf irst can be directly related to the damping ratio Q44. Eq. (5.23) reduces to
Q44≈Qf irst. (5.25)
An histogram of the relative sensitivity of the first peak to the axial shear coefficient C44, computed from the specimens dimensions and estimated stiffness, is shown on figure 5.5. The sensitivity is larger than 0.8 for the majority of the population, which justify the approximation (5.25).
0.2 0.4 0.6 0.8 1
Figure 5.5: relative sensitvity of the first resonant mode to the axial shear stiffness coefficient C44. The sensitivity is larger that 0.8 for the majority of the specimens, which justify the approximation Q44≈Qf irst.
The main advantage of that simple analysis is that a damping ratio is directly estimated from a single measured peak width. As the first resonant peak is usually strong and isolated from the following peaks (see Figure 5.3), its width is measured easily and with good precision.
Anisotropic analysis of some selected specimens
Some specimens were selected for an anisotropic analysis of the viscoleaticity based on the inversion of Eq. (5.23). The criterion for selection was that several resonant peaks were clearly marked and separated from the other, and that those peaks could be unambiguously associated to their predicted counterpart. The specimen corresponding to the spectra in Fig. 5.3is a good example: the four first peaks are not close to each other, their peaks are very well defined and their width can hence be measured with good precision.
For these specimens, the damping ratios were inverted using the analogous of Eq. (5.23) formulated for the Young’s moduli, shear moduli, and Poisson’s ratio.
Moreover, as RUS is intrinsically less sensitive to the Poisson’s ratios than to other engineering moduli (Chapter3) and because the imaginary part of the Poisson’s ra-tios is expected to be small, their imaginary part was neglected in the inversion. This reduces the number of parameters to three, as the relation between the transverse shear modulus C66, Young’s modulusE1 and Poisson’s ratio ν12
C66= E1
2(1 +ν12) (5.26)
entails the equality of the damping ratios Q66 and QE1 when ν12 is real. Then Q−1k = 2E3
5.3. Results 101
4.5 4.6 4.7 C66 6.4 6.5 6.6 6.7
C44 C44
C44
7 8 9 10 11
C13
C13 C13
30 32 34
C22 15 16 17 18 19
C11
C11
C22 C66
C22
Figure 5.6: 1D marginal posterior distribution of the elastic coefficients for a given specimen (diagonal), obtained using kernel density estimation, and 2D scattering plots showing the posterior correlations. The maxima of the marginal distribution are indi-cated by crosses and the 68%confidence intervale represented by horizontal lines.
Reduction to a three-parameters model drastically improves the conditioning of the inversion, even for a low number of measured peak width.
5.3 Results
Between 9 and 21 frequencies were obtained form the measured spectra for each specimen, with an average of 16 frequencies per specimen. The first resonant fre-quency of each specimen was in the range 100−400 kHz and the highest frefre-quency in the range 200 kHz−1 MHz.
Typical posterior marginal distributions and 2D scatter plots obtained for a specimen from the sampled distribution are plotted on figure 5.6. This representa-tion allows to visualize both the variance and covariances of the coefficients. The majority of the posterior distributions were clearly uni-modal, as on figure5.6, but some were more complex and multimodal. The posterior distribution presented on figure 5.7 was, subjectively, the worst looking distribution. However, in all cases the positions of the maximum a posteriori estimates were clearly identified. Such representation are also available for the other elastic parameters (Young’s modulus, Poisson’s ratios).
3.6 3.8 4 4.2 4.4 4.6 C66 C66
5.6 5.8 6 6.2 6.4 6.6 C44 C44
5 10 15
C13
C13 C13
25 30 35
C22
15 20 25
C11
C22
C11
C13
C44 C22
Figure 5.7: Same as Fig.5.6above, but for another specimen. In that particular case the distribution has two modes.
The median values of the relative half-width of the68%confidence interval were 7.3, 4.4, 10.9, 1.3, and 2.0 % for C11,C33, C13,C44, and C66, respectively and 2.3, 2.8, 10.1, 7.6, 8.4 %for E1,E3,ν12,ν13, and ν31, respectively.
For 4 specimens, the sampled values ofσwere very high, about 0.05, correspond-ing to an error of 5% between measured and predicted frequencies. This is much higher that what should be obtained in a successful experiment. These specimens were excluded of all the subsequent analysis. For the other specimens, the error is about 1%. The median value of the MAP estimate of σ was 0.83%. It reflects the relative agreement between the predicted and measured frequencies.
The estimated stiffness coefficients and Young’s moduli, as well as the Poisson’s ratios, are plotted as a function of bone density in figure 5.8. The anisotropy ratios are plotted on figure5.9. The range of apparent mass density was 1.58 - 2.01 g/cm3. Additionally, the mean values of the Poisson’s ratios are compared to results of the literature in Table 5.2.
The damping ratio of the axial shear coefficient is plotted on figure 5.10 as a function of density, while results of the anisotropic viscoelastic analysis for seven selected specimens are presented in Fig.5.11. From 7 to 9 peak widths, the damping ratios for the shear and Young’s moduli were estimated. The predicted peak width agreed with measurement within 5% in average (Fig. 5.12), which is comparable to the precision of the measurement. Despite the somewhat large uncertainties
5.3. Results 103
Figure 5.8: Longitudinal stiffness (C11 and C33; top left), shear stiffness (C44 and C6; top rigth), Young’s moduli (E1andE3; bottom left), and Poisson’s ratios (ν12,ν13
andν31; bottom rigth) as function of apparent bone mass density.
1.6 1.7 1.8 1.9 2
Figure 5.9: Anisotropy ratios as functions of apparent bone density.
1.6 1.7 1.8 1.9 2 15
20 25 30 35
Bone density Q firstpeak≈ Q 44
R = 0.84, p < 10−5
Figure 5.10: Damping factor of the first resonant peak, approximetly equal to the axial shear damping ratio Q44, as a function of bone mass density.
on the results, anisotropy of the damping ratios can be observed: Q−1E
3 < Q−144 <
Q−1E
1 =Q−166, i.e. the damping is weaker in axial direction of bone, both for Young’s and shear moduli. Adding the Poisson’s ratios in the analysis did not significantly improve the agreement, as represented on Fig. 5.12, which shows that their effect on the peak widths was indeed negligible.