Cécile Baron

Effect of porosity on effective diagonal stiffness coefficients (cii) and elastic anisotropy of cortical bone at 1MHz: A finite-difference time domain study

Finite-difference time domain (FDTD) numerical simulations coupled to real experimental data were used to investigate the propagation of 1 MHz pure bulk wave propagation through models of cortical bone microstructures. Bone microstructures were reconstructed from three-dimensional high resolution synchrotron radiation microcomputed tomography (SR-muCT) data sets. Because the bone matrix elastic properties were incompletely documented, several assumptions were made. Four built-in bone matrix models characterized by four different anisotropy ratios but the same Poissons ratios were tested. Combining them with the reconstructed microstructures in the FDTD computations, effective stiffness coefficients were derived from simulated bulk-wave velocity measurements. For all the models, all the effective compression and shear bulk wave velocities were found to decrease when porosity increases. However, the trend was weaker in the axial direction compared to the transverse directions, contributing to the increase of the effective anisotropy. On the other hand, it was shown that the initial Poissons ratio value may substantially affect the variations of the effective stiffness coefficients. The present study can be used to elaborate sophisticated macroscopic computational bone models incorporating realistic CT-based macroscopic bone structures and effective elastic properties derived from muCT-based FDTD simulations including the cortical porosity effect.

Propagation of elastic waves in a fluid-loaded anisotropic functionally graded waveguide: Application to ultrasound characterization

Non-destructive evaluation of heterogeneous materials is of major interest not only in industrial but also in biomedical fields. In this work, the studied structure is a three-layered one: A laterally heterogeneous anisotropic solid layer is sandwiched between two acoustic fluids. An original method is proposed to solve the wave equation in such a structure without using a multilayered model for the plate. This method is based on an analytical solution, the matricant, explicitly expressed under the Peano series expansion form. This approach is validated for the study of a fluid-loaded anisotropic and homogeneous plane waveguide with two different fluids on each side. Then, original results are given on the propagation of elastic waves in an asymmetrically fluid-loaded waveguide with laterally varying properties. This configuration notably corresponds to the axial transmission technique to the ultrasound characterization of cortical bone in vivo.

Ratio between mature and immature enzymatic cross-links correlates with post-yield cortical bone behavior: An insight into greenstick fractures of the child fibula

As a determinant of skeletal fragility, the organic matrix is responsible for the post-yield and creep behavior of bone and for its toughness, while the mineral apatite acts on stiffness. Specific to the fibula and ulna in children, greenstick fractures show a plastic in vivo mechanical behavior before bone fracture. During growth, the immature form of collagen enzymatic cross-links gradually decreases, to be replaced by the mature form until adolescence, subsequently remaining constant throughout adult life. However, the link between the cortical bone organic matrix and greenstick fractures in children remains to be explored. Here, we sought to determine: 1) whether plastic bending fractures can occur in vitro, by testing cortical bone samples from childrens fibula and 2) whether the post-yield behavior (ωp plastic energy) of cortical bone before fracture is related to total quantity of the collagen matrix, or to the quantity of mature and immature enzymatic cross-links and the quantity of non-enzymatic cross-links. We used a two-step approach; first, a 3-point microbending device tested 22 fibula machined bone samples from 7 children and 3 elderly adults until fracture. Second, biochemical analysis by HPLC was performed on the sample fragments. When pooling two groups of donors, children and elderly adults, results show a rank correlation between total energy dissipated before fracture and age and a linear correlation between plastic energy dissipated before fracture and ratio of immature/mature cross-links. A collagen matrix with more immature cross-links (i.e. a higher immature/mature cross-link ratio) is more likely to plastically deform before fracture. We conclude that this ratio in the sub-nanostructure of the organic matrix in cortical bone from the fibula may go some way towards explaining the variance in post-yield behavior. From a clinical point of view, therefore, our results provide a potential explanation of the presence of greenstick fractures in children.

Using the gradient of human cortical bone properties to determine age-related bone changes via ultrasonic guided waves.

Bone fragility depends not only on bone mass but also on bone quality (structure and material). To accurately evaluate fracture risk or propose therapeutic treatment, clinicians need a criterion, which reflects the determinants of bone strength: geometry, structure and material. In human long bone, the changes due to aging, accentuated by osteoporosis are often revealed through the trabecularization of cortical bone, i.e., increased porosity of endosteal bone inducing a thinning of the cortex. Consequently, the intracortical porosity gradient corresponding to the spatial variation in porosity across the cortical thickness is representative of loss of mass, changes in geometry (thinning) and variations in structure (porosity). This article examines the gradient of material properties and its age-related evolution as a relevant parameter to assess bone geometry, structure and material. By applying a homogenization process, cortical bone can be considered as an anisotropic functionally graded material with variations in material properties. A semi-analytical method based on the sextic Stroh formalism is proposed to solve the wave equation in an anisotropic functionally graded waveguide for two geometries, a plate and a tube, without using a multilayered model to represent the structure. This method provides an analytical solution called the matricant and explicitly expressed under the Peano series expansion form. Our findings indicate that ultrasonic guided waves are sensitive to the age-related evolution of realistic gradients in human bone properties across the cortical thickness and have their place in a multimodal clinical protocol.

Analyzing the anisotropic Hooke's law for children's cortical bone.

Child cortical bone tissue is rarely studied because of the difficulty of obtaining samples. Yet the preparation and ultrasonic characterization of the small samples available, while challenging, is one of the most promising ways of obtaining information on the mechanical behavior of non-pathological children׳s bone. We investigated children׳s cortical bone obtained from chirurgical waste. 22 fibula or femur samples from 21 children (1-18 years old, mean age: 9.7±5.8 years old) were compared to 16 fibula samples from 16 elderly patients (50-95 years old, mean age: 76.2±13.5 years old). Stiffness coefficients were evaluated via an ultrasonic method and anisotropy ratios were calculated as the ratio of C33/C11, C33/C22 and C11/C22. Stiffness coefficients were highly correlated with age in children (R>0.56, p<0.01). No significant difference was found between C11 and C22 for either adult or child bone (p>0.5), nor between C44 and C55 (p>0.5). We observe a transverse isotropy with C33>C22=C11>C44C55>C66. For both groups, we found no correlation between age and anisotropy ratios. This study offers the first complete analysis of stiffness coefficients in the three orthogonal bone axes in children, giving some indication of how bone anisotropy is related to age. Future perspectives include studying the effect of the structure and composition of bone on its mechanical behavior.

Characterisation of the difference in fracture mechanics between children and adult cortical bone

Clinical literature describes a specific type of children bone fracture, known as “greenstick facture” which is never encountered for adult bone. Concerning children bone, there is a tremendous lack of mechanical references. Indeed, the few studies which explored the mechanical characteristics of growing process in bone dealt with samples close to cancerous cells [1], or with samples from cadavers [2]. These studies gave dispersive results and did not provide insights to the two different kinds of fracture (i.e. brittle for mature bone and plastic for growing bone). Part of the answer could lie in the evolution of the biochemical composition of cortical bone; indeed, Bala et al. [3] have shown that the elasticity depends on the mineral part of the bone matrix and the plasticity on the organic part (collagen 1). This organic part of cortical bone seems to differ between adult and children. Saito et al. [4] have shown that the main non enzymatic crosslinks in mature bone (PYD+DPD) are different from those of growing bone (DHLNL+HLNL). It seems to us that the difference between plastic growing bone fractures and brittle adult bone fractures could be explained by this difference in the non enzymatic collagen crosslinking. We performed three point microbending tests on children and adult bone to evaluate the mechanical Young’s modulus (Em) and the plastic strain energy (ωp). The results are in agreement with the clinical observations. The goal of this study is to explain these differences in mechanical behaviour between children and adult bone by using a biochemical analysis of the organic part quantifying the composition of the collagen.

Investigation of the porous network as a determinant of the overall stiffness of cortical bone: Mori‐Tanaka model vs. ultrasound propagation

Assessing the effect of porosity on stiffness in cortical bone remains an important issue that has already been addressed with several models. The originality of the present work is to compare two models of cortical bone: one uses a realistic porous network (voxel 20 microns) reconstructed from synchrotron radiation tomography; the other considers cylindrical pores aligned in a single direction. In the first case, overall elastic properties are evaluated indirectly by means of finite difference time domain simulation of ultrasound bulk wave propagation at 1 MHz. In the second model, effective elasticity is calculated by means of a Mori‐Tanaka scheme based on Eshelby solution for cylindrical inclusions with ellipsoidal cross section. Overall properties were evaluated with the two methods for 18 porosity values, each corresponding to a reconstructed bone volume. The diagonal stiffness coefficients of the overall bone material estimated with the two methods compared well. Results for the stiffness coefficien...

Measuring mass density and ultrasonic wave velocity: A wavelet-based method applied in ultrasonic reflection mode

When assessing ultrasonic measurements of material parameters, the signal processing is an important part of the inverse problem. Measurements of thickness, ultrasonic wave velocity and mass density are required for such assessments. This study investigates the feasibility and the robustness of a wavelet-based processing (WBP) method based on a Jaffard-Meyer algorithm for calculating these parameters simultaneously and independently, using one single ultrasonic signal in the reflection mode. The appropriate transmitted incident wave, correlated with the mathematical properties of the wavelet decomposition, was determined using a adapted identification procedure to build a mathematically equivalent model for the electro-acoustic system. The method was tested on three groups of samples (polyurethane resin, bone and wood) using one 1-MHz transducer. For thickness and velocity measurements, the WBP method gave a relative error lower than 1.5%. The relative errors in the mass density measurements ranged between 0.70% and 2.59%. Despite discrepancies between manufactured and biological samples, the results obtained on the three groups of samples using the WBP method in the reflection mode were remarkably consistent, indicating that it is a reliable and efficient means of simultaneously assessing the thickness and the velocity of the ultrasonic wave propagating in the medium, and the apparent mass density of material.

Influence of age and localisation on pedicle fixation in immature porcine spines

Pedicle screw is a widely used anchorage system for spinal devices for surgical treatments of paediatric deformities. Its fixation depends on various parameters such as bone quality and the screw-bone interface (Gao et al. 2011). During the development of a new spinal system, animal experiments may be performed for first validation, especially on the porcine model, well adapted because it matches best with the pediatric spine in terms of size and shape of the vertebrae (Yazici et al. 2006). However, in the literature, there are few data concerning pull-out loads of pedicle screws inserted in young pigs, and to the best of the knowledge of the authors, there is no study comparing pull-out loads of pedicle screw inserted in porcine spines according to the age of the animal. The current study was performed (1) to obtain ultimate pull-out strengths of pedicle screws in young pigs of two different ages, (2) to assess the effect of the localisation of the vertebrae (lumbar or thoracic) and (3) to assess the effect of the age of the animals on those ultimate pull-out forces.

Thin bone sample assessment using ultrasonic transmitted signals based on wavelet processing method

The wavelet-based processing (WBP) method based on Meyer-Jaffard algorithm was implemented to simultaneously determine the thickness and velocity of thin cortical bone samples. Two groups of bovine samples were measured by one pair of immersion transducers with nominal frequency 2.25MHz. The WBP method was used to estimate the times of flight (TOF) of two pulses contained by one transmitted signal. The mean relative error of thickness measurement is 6.13%. The mean velocities and their standard deviation for two different groups are 3399 ± 131 m/s and 3502 ± 182 m/s, which lead to the average relative errors from the pulse-mode method of 8.38% and 11.15% respectively. The results demonstrate that the WBP method is able to measure thin bone samples whose thickness is comparable or even less than ultrasound wavelength and provides the potential to assess more reliable initial models for the further imaging process.