Simulation of the errors due to imperfect specimen geometry
In RUS, the inverse problem to determine stiﬀness constants is solved assuming that the specimen is a perfect RP. In this section, we investigate the uncertainty 30 Chapter 2.
Figure 2.9: The mean and 95% confidence intervals of the errors on stiﬀness constants and engineering moduli corresponding to case (D), (F), (D+F) and (S). The error bars show the upper and lower bounds of the intervals and the mean values are represented at the center of the errorbars by the ’circle’ or ’square’ makers. For case (D), (F) and (D+F) the intervals were estimated as mean ± 2×SD, for case (S) they were evaluated by fitting the cumulative distribution functions of the errors using kernel density estimators.
on stiﬀness associated to this assumption resorting to a ’virtual’ RUS experiment (Fig. 2.8 (block S) and Fig. 2.10):
(1) For each of the 23 bone specimens, the resonant frequencies ffem were cal-culated using the finite element method considering the actual specimen’s geometry derived from SR-µCT images, measured mass mexp and specimen’s stiﬀness Cexpij determined in the usual manner assuming a perfect RP shape. Details on the finite element implementation are given in appendix (Appendix A).
(2) The stiﬀness constants Cfemij of each specimen were estimated solving the inverse problem defined by the frequencies ffem (the first 40 frequencies) considered as measurements and a forward model characterized by a perfect RP geometry (dimensions dimexp) and specimen’s mass (mexp) (Sec. 2.3).
These resulting Cfemij are the stiﬀness constants of a RP bone specimen that would exhibit the same resonant frequencies as the imperfect shape bone specimens with stiﬀness constants Cexpij. Constants Cfemij are biased by imperfect specimen geometry and are compared to the true stiﬀness constants of the specimen used in exp fem Cijfem−Cijexp the FEM model (Cij ). Namely, we calculate the errors δCij = × Cijexp 100%.
The errors on stiﬀness constants and the engineering moduli due to imperfect ge-ometry of the specimens are summarized in Table 2.2 (last column). As only 23 specimens were included and the errors were not normally distributed, the 95% CIs of the errors (Table 2.3 and Fig. 2.9) were evaluated by fitting the cumulative dis-(Section 2.3.2), ffem are the resonant frequencies calculated from the actual specimen geometry, dimexp are the dimensions of the specimens measured by caliper, Cfemij are the stiﬀness constants calculated by solving the inverse problem, and δCfemij represent the estimation errors.
tribution functions of the errors using kernel density estimators. For all the stiﬀness constants, there is a bias, i.e. the mean value of the errors is not zero and it can be positive or negative depending on the constant (the mean values vary from -3.19% to 3.56%). The SD of the errors varies from 0.68% to 2.08%. In particular, the errors on shear stiﬀness constants present a smaller variation than longitudinal and oﬀ-diagonal ones (see the 95% CIs in Table 2.3) and the errors on Young’s mod-uli present slightly less variability compared to the longitudinal stiﬀness constants (Table 2.2 and 2.3).
Discussion and conclusion
In this study, we performed simulations to quantify the errors on the stiﬀness con-stants determined from RUS measurements. We used typical elasticity values of human cortical bone as reference and studied the eﬀects of errors due to (1) uncer-tainties on the measurement of frequencies; (2) uncertainties on the measurement of dimensions (assuming a perfect RP shape); (3) imperfect specimen’s geometry (deviation from a perfect RP). The first two points were addressed with a calcu-lation of error propagation with Monte-Carlo simulations and the third point was addressed using actual experimental data on a collection of 23 bone specimens. The main parameters of the Monte-Carlo simulations were the assumed level of error on experimentally determined resonant frequencies, set to 0.5%, and experimentally determined specimen’s dimensions, set to 0.04 mm (∼1%). The choice of these values is consistent with our experience of using RUS to measure bone specimens (Bernard et al., 2013).
Using micro-CT, we could quantify the range of geometrical errors associated to a simple specimen’s preparation procedure. We found that perpendicularity and parallelism errors were in average less than 1◦ and always less than 2◦ (Fig. 2.6).
Overall, we found errors on elasticity values of a few percents, or less than one percent, depending on the considered stiﬀness constant. Note that we discuss the accuracy errors reporting the 95% CIs of the error. Consistent with the findings of several previous studies (e.g., (Migliori and Sarrao, 1997; Landa et al., 2009)) we found that the oﬀ-diagonal stiﬀness constants presented the highest errors and shear constants the smallest ones. This is related to the higher sensitivity of RUS to shear stiﬀness constants. For all the stiﬀness constants but C13, dimension uncer-tainties lead to larger errors in elasticity compared to the case where only frequency uncertainties are considered (Table 2.2 and Fig. 2.9). For C13, the largest error is observed for frequency uncertainties, suggesting that C13 may be less sensitive to di-mension imperfections than to resonant frequencies in current simulation conditions. Interestingly, dimension uncertainties or a coupling of frequency and dimension un-certainties caused similar levels of errors on the stiﬀness constants (Table 2.2 and Fig. 2.9).
Deviation of the actual specimen’s shape from a perfect RP aﬀects the accuracy of the stiﬀness constants measured with RUS. This is because the forward model used to solve the inverse problem, assuming a perfect RP geometry, is not correct. The approach introduced in Section 2.5 aimed at simulating the eﬀect of this source of error. It is important to note that, in general (when a micro-CT scan of the spec-imen is not available), only the mass can be accurately measured as opposed to the dimensions (because the geometry is in general not perfect). This is the reason why mass (mexp) but not mass density was used in the simulations in Secs. 2.4 and 2.5. We have observed that most of the caliper-measured volumes were overestimated of approximately 3% in average compared to the volumes deduced from SR-µCT images. Accordingly, the quantified elasticity errors are a result of both overesti-mated dimensions and irregularity of the RP shape. The elasticity errors due to an imperfect RP geometry (Table 2.3 and Fig 2.9) were between 2.3%∼6.2%. Overall, the errors due to an imperfect RP geometry are found to be of the same order as the errors calculated by Monte-Carlo simulations due to frequency or dimension uncertainties.
It is noteworthy that the values of σf obtained from simulations in the present study (σf ≈ 0.58% with Monte-Carlo simulations and σf ≈ 0.29% using FEM sim-ulations with the imperfect shape) are similar to values reported for actual RUS measurements of bone and other attenuative materials (Bernard et al., 2013, 2014) where σf is typically in the range 0.25-0.40%. This suggests that the simulations accurately reproduce the experimental error characteristic of RUS measurements. The level of errors quantified in the present study are consistent with the reported precision of RUS for human cortical bone application (3%, 5% and 0.4% for lon-gitudinal, oﬀ-diagonal and shear stiﬀness constants, respectively) (Bernard et al., 2013), estimated from the RMSE σf .
This study has introduced an original methodology to quantify errors in RUS measurements. The method was applied to bone but could be used to assess the accuracy for RUS measurements of various materials. Note that it has been pos-sible to implement Monte-Carlo simulations only because an automated pairing of frequencies (for the calculation of the objective function) was possible. This auto-mated pairing was initially developed to process spectra of attenuative materials where several resonant peaks can not be retrieved (Bernard et al., 2015) and it is also eﬃcient to process synthetic resonant frequencies as in the present study where no peak is missing.
This study has some limitations. We used simulated resonant frequencies as proxy for RUS data as input to the inverse problem. Precisely, the eigenfrequencies of the first forty vibration modes were used. In actual RUS experiments to measure bone, a maximum of fifteen to twenty frequencies among the first forty can actually be retrieved due to peak overlapping (Bernard et al., 2014). In theory, taking into account more frequencies should improve the precision of the determination of stiﬀness constants because more information is used for the inverse problem. However, in practice, the achievable precision also depends on the quality of the frequency measurement which decreases in the higer frequency range due to the increased modal density and peak overlapping. Since the resonant frequencies are much more sensitive to shear stiﬀness constants (Zadler et al., 2004), it is expected that using less frequencies than in the present study would essentially decrease the precision of constants C11, C33 and C13 but would have little impact on the precision of the shear stiﬀness constants. The results of the simulation in Sec. 2.5 critically rely on the actual pixel size in SR-µCT experiments, because the exact shape of the specimens were used to compute the ’true’ resonant frequencies for the inverse problem. However, we did not perform calibration for identifying the actual pixel size during SR-µCT experiments. This could partly aﬀect or bias the results.
Another limitation is that we did not simulate the error on stiﬀness constants due to a combination of frequency uncertainty and imperfect RP geometry. In view of the results of Sec. 2.4, we expect that elasticity errors would only be slightly larger. Furthermore, some sources of errors in RUS have not been considered such as the eﬀect of imperfect boundary conditions (Yoneda, 2002) and the uncertainty on the measurement of specimen’s mass.
The validation of the measurement of bone elasticity with RUS relies (1) on the successful measurement of a reference transverse isotropic material with a Q-factor similar as bone’s Q-factor (Bernard et al., 2014); (2) on the comparison of the stiﬀness constants obtained with RUS and from the independent measurement of the time-of-flight of shear and longitudinal waves in bone specimens (Bernard et al., 2013; Peralta et al., 2017); and (3) on the results of the present study focused on the quantification of accuracy errors. The latter suggest that despite the typical non-perfect geometry of bone specimens and despite the relatively large uncertainty in the measurement of the bone resonance frequencies (due to attenuation), the stiﬀness constants are obtained with a maximum error of a few percents. A very conservative accuracy value can be quantified by the larger absolute value of the (non symetric) 95% CI bounds; accuracy defined like this was 6.2% for longitudinal stiﬀness and 3.3% for shear stiﬀness, 5.1% for Young’s moduli and 5.6% for Poisson’s ratios (Table 2.3).
To further enhance the accuracy of bone RUS measurement, possible paths would be (1) using a specific implementation of the Rayleigh-Ritz method for non-rectangular parallelepiped specimen (Landa et al., 2009), provided that the angles between the specimen’s surfaces can be measured; (2) decreasing the frequency uncertainty by improving the signal processing of RUS spectra.
This work has received financial support from the Agency National Research un-der the ANR-13-BS09-0006 MULTIPS and ANR-14-CE35-0030-01 TaCo-Sound projects and was done in the framework of LabEx PRIMES (ANR-11-LABX-0063) of Université de Lyon. The authors would like to thank Rémy Gauthier, Hélène Follet and David Mitton for the collection of bone specimens and the help in con-ducting SR-µCT imaging experiments. The authors wish to acknowledge Didier Cassereau for providing technical support with numerical computations and Pascal Dargent for designing the setup and the protocol of bone specimen preparation, as well as the ESRF for providing beamtime through the experiment MD927 and Lukas Helfen for his assistance in image acquisition on beamline ID19.
This chapter is a research article prepared for submission but not yet submitted at the time of writing this thesis. This chapter presents a study about the relationships between microstructural, compositional properties and cortical bone stiﬀness. An exhaustive amount of bone material characteristics were quantitatively measured by state-of-the-art techniques, including RUS, synchrotron radiation micro-computed tomography, Fourier transform infrared microspectroscopy and biochemistry exper-iments with the objective to find the interplay between them.
Human cortical bone is a complex composite material composed of about 70% mineral (hydroxyapatite), 22% proteins (type I collagen) and 8% water by weight (Augat and Schorlemmer, 2006), assembled in a hierarchical structure that extends over several organization levels (Currey, 1990; Fratzl and Weinkamer, 2007). At the nanoscale (nanometer-scale), the fibrils mainly constituted by collagen and hy-droxyapatite are arranged in fibers. Mineralized fibers are arranged to form bone lamellae whose typical thickness is about several micrometers. Within lamellar sublayers, the orientation of the fibers may vary and presents a twisted plywood structure (Giraud-Guille, 1988). At the microscale (micrometer-scale), the osteon, a cylindrical structure organized by several concentric lamellae around a Haversian canal (20-100 µm in diameter), constitutes the basic structural unit at this level. Connected by Volkmann’s cannals (several tens µm in diameter), the Haversian canals form the microstructure of bone or the so-called porous network with the presence of resorption cavities and osteocyte lacunae and canaliculi (a few µm to less than 1 µm in diameter). As the pores mainly contain fluids and soft tissues (e.g., blood vessels and nerves), human cortical bone is often described as a porous composite material consisting of a soft organic matrix hardened by a mineral phase (Fritsch and Hellmich, 2007; Deuerling et al., 2009; Grimal et al., 2011b; Parnell et al., 2012; Granke et al., 2015).
As the main organ who carries most of the mechanical loading of human body, bone undergoes a permanent biological remodelling process regulated by mechanosensitive osteocytes, which allows it to adapt to the mechanical load (Klein-Nulend et al., 2003). Mechanosensitive osteocytes are capable of sensing the local strain determined by bone stiﬀness for a given load. Hence investigating bone stiﬀness in detail should improve the understanding of bone functional adaptation mechanisms and bone strength. For instance, significant correlations were found between bone stiﬀness and strength in several mammals, such as human, bovine and deer (Fyhrie and Vashishth, 2000). In particular, cortical bone stiﬀness at the mesoscale (millimeter-scale) is of special interest as it has a direct impact on the mechanical behavior of bone at the macroscale level (Rho et al., 1998; Gri-mal et al., 2011b) at which cortical bone acts, in concert with the overall gross shape of a bone, to resist to functional loads (Currey, 2002). The mesoscopic level is also appropriate to investigate the regional variations of the elastic properties within a bone (Rohrbach et al., 2015), which is necessary to refine finite element models to predict patterns of stress and strain of proximal femur (Zysset et al., 2015; Liebl et al., 2015), of vertebra (Keaveny et al., 2007; Pahr et al., 2014; Zysset et al., 2015) and distal radius (Engelke et al., 2016; Kawalilak et al., 2016; Christen et al., 2013), the three major skeletal sites of fracture for osteoporotic patients. The intra-individual (Orías et al., 2009) and inter-individual (Rudy et al., 2011; Bernard et al., 2016) variations of mesoscopic stiﬀness are the consequences of the remodeling process and the structure-function adaptation mechanisms of bone, as well. As mesoscopic stiﬀness depends on both microstructure and tissue properties at all the smaller length scales, a clear understanding of the compositional and microstructural variables that govern bone mesoscopic stiﬀness variations would help in better understanding the remodeling process or bone functional adaptation mechanisms, as well as developing more accurate mechanical models (Deuerling et al., 2009; Fritsch and Hellmich, 2007).
The inquiry of the role of compositional and microstructural variables in the variation of bone stiﬀness dates back to the 70’s (Currey, 1975; Carter and Hayes, 1977) since when the importance of porosity and mineral content have been identi-fied. Although vascular porosity has been found to be an important determinant of bone mesocale stiﬀness (Currey, 1975; Dong and Guo, 2004; Granke et al., 2011), the role of the 3D microstruture of the pore network beyond the alteration of poros-ity in cortical bone stiﬀness is barely studied. The microstructural features, such as pore size, pore number, pore diameter, pore connectivity are the footprints of bone remodeling process and may also have an eﬀect on the alteration of bone stiﬀness. A recent study has demonstrated that changes in pore network micro-architecture may impact fibula cortical bone stiﬀness during growth and aging (Bala et al., 2016). On the other hand, the properties of extracellular bone matrix also have an eﬀect on bone mesoscale stiﬀness, all the more because it occupies most of the cortical bone volume. An important property of bone matrix is the mineral content which has been evaluated in various ways, including chemically dissolved (Currey, 1988), burning up the organics in high temperature (Öhman et al., 2011), and inferred by X-ray based methods (Follet et al., 2011; Nuzzo et al., 2002b). Depending on the evaluation methods diﬀerent names were used to describe the same quantity, such as calcium content, ash density (weight of the ashed mineral material after burning divided by specimen volume), bone mineral density and degree of mineralization of bone (DMB). The Young’s modulus of bone was found to be correlated with the calcium content (Currey, 1988) and the variation of the Young’s modulus between children and adults’ cortical bone can be largely explained by the ash density (Öh-man et al., 2011). However, DMB which may be able to be measured clinically was not correlated with bone elasticity at the mesoscale (Follet et al., 2011) and there is a lack of studies to correlate the volumetric DMB values with bone elasticity.
Although several studies investigating bone stiﬀness determinants have involved many microstructural and compositional (mineralization, crystallinity and collagen) features, bone was seldom considered as an anisotropic material and the relative contribution of these aspects to bone anisotropic stiﬀness were not quantified. In this work, to address these issues, we carry out a thorough investigation of the determinants of cortical bone mesoscale stiﬀness, including both the microstructural and compositional aspects and their relative contribution using the same subjects and by applying several state-of-the-art experimental modalities. Specifically, the anisotropic bone stiﬀness, bone microstructure and DMB were precisely measured by the latest techniques. Other compositional properties such as crystallinity which reflects the size and perfection of crystals, maturity of mineral and collagen, as well as collagen crosslinks were also quantified because they are also a vital part of bone composition and were also associated with bone stiﬀness before (Banse et al., 2002; Oxlund et al., 1995), especially in pathological bone, such as osteogenesis imperfecta (Vanleene et al., 2012) or drug treated bone, e.g., by bisphosphonates (Bala et al., 2012; Ma et al., 2017).
Materials and methods
Bone specimens were harvested from the left femur of 29 human cadavers. The femurs were provided by the Départment Universitaire d’Anatomie Rockefeller (Lyon, France) through the French program on voluntary corpse donation to sci-ence. Among the donors, 16 were females and 13 were males (50 − 95 years old, 77.8 ± 11.4, mean±SD). The fresh material was frozen and stored at −20◦C. The samples were slowly thawed and then, for each femur, approximately a 40 mm thick cross section was cut perpendicular to the bone axis from the mid-diaphysis. The cross section was separated to four sections (lateral, medial, anterior and posterior).
In each of the lateral and medial anatomical quadrants, 4 rectangular parallelepiped shaped specimens (set #1, #2, #3 and #4) were prepared along the axial direction for diﬀerent testing purposes (Fig. 5.1). The nominal specimen size of set #1, #2, #3 and #4 was 3×4×5 mm3, 3×4×25 mm3, 3×4×0.5 mm3 and 3×4×5 mm3 in radial (axis 1), circumferential (axis 2) and axial direction (axis 3), respectively, de-fined by the anatomic shape of the femoral diaphysis. One subject with a porosity higher than 30% was not considered for the measurements and was discarded. As part of the experiments failed on specimens from 2 subjects, they were also excluded which finally led to 26 subjects (52 specimens) for each set (#1, #2, #3 and #4) of specimens. All specimens were kept hydrated during sample preparation.
Bone stiﬀness measurement
The stiﬀness constants of bone specimens (set #1 and #4) were measured by res-onant ultrasound spectroscopy (RUS) following the procedure previously described (Bernard et al., 2013) which consists of the following steps: (1) A bone specimen was placed on two opposite corners between two ultrasonic transducers (V154RM, Panametrics, Waltham, MA), one for emission and one for reception, to achieve a free boundary condition for vibration. (2) The frequency response of the vibration in a specified bandwidth was amplified by a broadband charge amplifier (HQA-15 M-10 T, Femto Messtechnik GmbH, Berlin, Germany) and then recorded by a vector network analyzer (Bode 100, Omicron Electronics GmbH, Klaus, Austria), from which between 20-30 first resonant frequencies were extracted. (3) Assuming a transversely isotropic symmetry (Yoon and Katz, 1976; Orías et al., 2009), the stiﬀness constants Cij (ij = 11, 33, 13, 44, 66) (Voigt notation), were automatically calculated by optimizing the misfit function between the experimental and model predicted resonant frequencies (inverse problem), which is formulated in a Bayesian framework (Bernard et al., 2015). The a prior information of the distribution of the stiﬀness constants required for the Bayesian analysis was taken from a previous study (Granke et al., 2011). In the elastic tensor, C12 = C11 − 2C66 and (1 − 2) is the isotropic plane; C11 and C33 are the compression stiﬀness constants, C12 and C13 are the oﬀ-diagonal stiﬀness constants and C44 and C66 represent the shear stiﬀness constants. The mass density of each specimen which is needed in stiﬀ-ness determination was derived from the average values of four mass (precision: 0.1 mg) and dimensions measurements (precision: 0.01 mm). The experimental errors, e.g. irregularity of specimen geometry and uncertainties of the extracted resonant frequencies, following this protocol, typically cause an accuracy error (standard deviation (SD)) of approximately 1.7% for the shear stiﬀness constants and approx-imately 3.1% for the compression and oﬀ-diagonal stiﬀness constants (Chapter 2).
After RUS measurements, bone specimens in set #1 were defatted for 12 hours in a chemical bath of diethylether and methanol (1:1) and rinsed in distilled water before SR-µCT scanning in order to comply with the local regulation by the European Synchrotron Radiation Facility (ESRF, Grenoble, France). This defatting protocol does not alter the anisotropic stiﬀness of human cortical bone measured by RUS as has been verified in a previous work (Chapter 6). Then, the specimens were scanned using SR-µCT 3D imaging, which was performed on the beamline ID19 at ESRF. This SR-µCT setup is based on a 3D parallel beam geometry acquisition (Salomé et al., 1999; Weitkamp et al., 2010). The beam energy was tuned to 26 keV by using a (Si111) double crystal monochromator. A full set of 2D radiographic images were recorded using a CDD detector (Gadox scintillator, optic lenses, 2048 × 2048 Frelon Camera) by rotating the specimen in 1999 steps within a 360◦ range of rotation. The detector system was fixed to get a pixel size of 6.5 µm in the recorded images in which a region of interest of 1400×940 pixels was selected to fit the specimen. For each specimen, the SR-µCT images were reconstructed.
In the reconstructed images of each specimen, a volume of interest (VOI, sized approximately 2.8 ×3.9 ×4.8 mm3) was selected manually for morphometric analy-sis. Following (Bala et al., 2016), the VOIs were binarized treating the void volumes as a solid and the bone phase as a background (Fig. 5.2). Then, the following mor-phometric parameters (Table 4.1) were calculated using the software CTAnalyser (V 1.16.1, Skyscan NV, Kontich, Belgium): pore volume fraction (φ), pore surface to pore volume ratio (PoS/PoV), the average diameter of the pores (PoDm), the average separation between pores (PoSp), pore number (PoN), connectivity density (ConnD) and pore pattern factor (PoPf), structure model index (SMI).
Materials and methods
the beamline ID 17 at ESRF. The configuration of the setup is similar to the afore-mentioned setup on beamline ID 19 but with a bit high energy 26.5 keV.
After the SR-µCT images were reconstructed. The gray levels of the images (Fig. 5.2), corresponding to the X-ray absorption coeﬃcients of bone, were cali-brated to estimate the 3D distribution of mineral content within a bone specimen. After the calibration, the gray levels were converted into DMB, i.e., the concentra-tion of bone mineral content (Nuzzo et al., 2002a). The reproducibility denoted as the coeﬃcient of variation (CV) of the DMB measured from the two experiments was evaluated to be 0.9% on the 23 specimens and the coeﬃcients of determination r2 was 0.76.
Fourier transform infrared microspectroscopy
The set #3 was measured by Fourier transform infrared microspectroscopy (FTIRM). The experimental procedure has been described in detail in Farlay et al.
(2010). From the spectrum, the following variables (Table 3.2) were retrieved: mineral-to-organ ratio (MinOrga) which represents the ratio of the mineral phase over the organic phase (collagenic and non-collagenic proteins), mineral maturity (MinMat) which reflects the transformation of non apatitic phosphates of the hy-drated layer into apatitic phosphates contained in the crystal core, carbonation (Carbon) which represents the quantity of carbonates incorporated into the bone mineral at a labile site and substituted in the apatite lattice due either to PO4 or OH, crystallinity index (CryInd) which corresponds to both crystalline domain size and how well the ions of the crystal are ordered in the unit cells and collagen maturity (CollMat). More details of the definition and mathematical calculation of these variables can be found in Table 3.2. The former four variables were grouped in the mineral variable group and the last one CollMat was grouped in collagen variable group.
The spared materials from the preparation of set #2 were cut into small pieces, powdered in liquid nitrogen-cooled freezer mil (Spex Centriprep, Metuchen, USA), and then demineralized by daily changing of 0.5 M EDTA in 0.05 M in Tris buﬀer, pH 7.4 for 96 hours at 4 ◦C. After removing the demineralized powder by washing with deionized water, the samples were suspended in phosphate buﬀered saline (0.15 M sodium chloride, 0.1 M sodium phosphate buﬀer, pH 7.4) and reduced in NaBH 4 at room temperature for 2 hours using a reagent/sample ratio of 1:30 (w/w). The reduced bone residues were then washed, freeze-dried and hydrolyzed in 6 M hydrochloric acid at 110 ◦C for 20 hours for the assessment of collagen and collagen cross-links.
Materials and methods
Normality of the variables were evaluated using Shapiro-Wilk test. One-way anal-ysis of variance (ANOVA) and Wilcoxon test (for the variables failed normality test) were performed to evaluate the diﬀerences of the data sets from lateral and medial anatomical quadrants. As some variables were not normally distributed, Spearman’s rank correlation coeﬃcients between stiﬀness constants and the mi-crostructural and compositional variables were calculated to highlight the potential explanatory variables of stiﬀness. The associated variables with any of the stiﬀness constants were kept and then, the relative contribution of the variables were quanti-fied by the Adjusted R2 (Adj-R2) and root-mean-square-error (RM SE) using single linear and stepwise multiple regression analyses. Note that all the variables except the stiﬀness constants were normalized between −1 and 1 using the equation, where x is the variable to be normalized, to highlight the relative importance of the explanatory variables in the model. Thereafter, the redundant variables in each group of variables (microstructure, mineral and collagen) were excluded. The re-maining variables from each group were gathered as the predictors to determine the best explanatory linear model of the stiﬀness constants. Note that for some variables which have significant diﬀerence between lateral and medial specimens, the multiple linear regression analyses were carried out both on the data set of lat-eral and medial specimens separately and the corresponding model are reported. If these variables were not retained as explanatory variables in the regression model, analyses were run again pooling the dataset from lateral and medial specimens. Finally, a two-variable model (φ and DMB, as detailed in Section 3.3.3) was eval-uated for each stiﬀness constant to test the hypothesis that φ and DMB are the most important determinants. Data were considered statistically significant for p < 0.05. Statistical analyses were made using the Matlab 2017a Statistics Toolbox (Mathworks Inc., Natick, MA, USA).
Table of contents :
1.1 Context and motivation
1.2 Hierarchical structure of bone
1.3 Importance of cortical bone and its elasticity
1.4 Assessment of cortical bone elasticity
1.4.1 Linear elasticity and Hooke’s law
1.4.2 Methods to measure bone mesoscopic elasticity
1.4.3 Methods to measure bone microelastic properties
1.5 MULTIPS project
1.6 Objectives of the thesis
1.7 Outline of the thesis
2 Quantification of stiffness measurement errors in resonant ultrasound spectroscopy of human cortical bone
2.2 RUS theory
2.3.2 Bone elasticity measurements by RUS
2.3.3 Specimen geometry
2.4 Simulation of the errors due to uncertainties on resonant frequencies and dimensions
2.5 Simulation of the errors due to imperfect specimen geometry
2.6 Discussion and conclusion
3 Relative contributions of microstructural and compositional properties to cortical bone stiffness at millimeter-scale
3.2 Materials and methods
3.2.2 Bone stiffness measurements
3.2.3 Bone microstructure
3.2.4 Degree of mineralization of bone
3.2.5 Fourier transform infrared microspectroscopy
3.2.6 Biochemical measurements
3.2.8 Micromechanics modeling
3.3.1 Descriptive statistics
3.3.2 Univariate correlation analysis
3.3.3 Multivariate regression model
3.3.4 Comparison between micromechanics model and experimental data
4 Role of microstructure on the anisotropic elasticity of human cortical bone
4.2 Materials and methods
4.2.2 Stiffness measurements with resonant ultrasound spectroscopy
4.2.3 Bone microstructure
4.2.4 Numerical model and method of solution
4.2.5 Model with idealized microsstructure
4.2.6 Calibration of bone matrix stiffness
4.2.7 Data analysis
4.3.1 Calibration of the model
4.3.2 Effect of pore shape
4.3.3 Descriptive statistics
4.3.4 Correlations between synthetic bone stiffness and microstructural variables
4.3.5 Effects of microstructure on bone stiffness
4.3.6 Effects of the simplified bone microstructure on bone stiffness
5 Bone tissue anisotropic elastic properties determined by an inverse homogenization method and resonant ultrasound spectroscopy
5.2 Materials and methods
5.2.2 Bone elasticity measurements
5.2.3 Bone microstructure
5.2.4 Degree of mineralization of bone
5.2.5 Forward problem
5.2.6 Inverse problem
5.2.7 Error propagation analysis
5.2.8 Microstrain on bone matrix
5.3.1 Experimental data and computation efficiency
5.3.2 Error propagation analysis
5.3.3 Comparison of the stiffness constants
5.3.4 Comparison between bone matrix elasticity and DMB
5.3.5 The variation of the microstrain
6 Cortical bone elasticity measured by resonant ultrasound spectroscopy is not altered by defatting and synchrotron X-ray imaging
6.2 Materials and methods
6.2.2 Elasticity measurements by resonant ultrasound spectroscopy
6.2.3 Bone defatting
6.2.4 Synchrotron radiation microtomography
6.2.5 Experimental Protocol
7 Assessment of trabecular bone tissue elasticity with resonant ultrasound spectroscopy
7.2.2 Image acquisition and processing
7.2.3 RUS measurements
7.2.4 μ-FE and model frequency calculation
8 Critical assessment of ultrasonic bulk wave velocity method in bone elasticity measurements
8.3.2 Ultrasonic velocity measurements
8.3.3 Resonant Ultrasound Spectroscopy
8.3.4 Data analysis
9 A comparison study of human cortical bone elasticity at multiple skeletal sites
9.2 Materials and method
9.2.2 Bone stiffness and porosity measurements