Accurate measurement of cortical bone elasticity tensor with resonant ultrasound spectroscopy

Asymptotic behavior of resonant frequencies in resonant ultrasound spectroscopy

Resonance ultrasound spectroscopy is a non-destructive technique used to assess materials' elastic and anelastic properties. It involves measuring the frequencies of free vibrations in a carefully prepared sample to extract material properties. In this paper, we investigate the asymptotic behavior of eigenfrequencies. Our primary focus is on analyzing the asymptotic behavior of eigenfrequencies, aiming to understand their rate of growth and convergence. We also make observations regarding the impact of elastic constants on eigenfrequencies.

Shear and longitudinal attenuations and quality factors of poly(methyl metacrylate) (PMMA) from 20 kHz to 12 MHz investigation with ultrasonic spectroscopy

PMMA is often considered as a calibration material for experimental benches dedicated to viscoelasticity investigation. Nevertheless, regarding literature, data concerning attenuation coefficients and quality factors are essentially available in the MHz frequency range and results in the low-frequency range are scarce and scattered. In this communication, thanks to the use of high-frequency ultrasonic spectroscopy between 2 and 8 MHz in the range 6 °C - 45 °C, Time-Temperature Superposition principle and Resonant Ultrasonic Spectroscopy (RUS), we show that both longitudinal and shear quality factors of PMMA decrease considerably for low frequencies (<MHz), and that the classically accepted linear laws describing attenuation as a function of frequency are valid only beyond several MHz. This variation is attributed to secondary relaxation processes such as γ relaxation considering the activation energy deduced from experimental data. Power laws are proposed to describe the evolution of quality factors and attenuation coefficients versus frequency in the 20 kHz - 12 MHz range.

Prediction of osteoporotic degradation of tibia human bone at trabecular scale

A theoretical numerical model is proposed to predict patient dependent osteoporotic bone degradation. The model parameters are identified through a particle swarm optimization algorithm and based on individual patient high resolution peripherical quantitative computer tomography (HRpQCT) scan data. The degradation model is based on cellular activity initiated by the elastic strain energy developed in the bone microstructure through patient's body weight. The macro (organ scale) and meso (trabecular scale) scale analyses are carried out and predicted bone volume fraction and microstructure evolution are compared with in-vivo experimental bone degradation for four elderly women over a period of 10 years. A significant correlation (r > 0.9) is observed between the model predictions and in-vivo experiments in all cases with an average deviation error of 1.46%. The model can easily be extended to other patients and provide good predictions for different population categories such as ethnicity, gender, age, etc.

Additively manufactured lattice structures with controlled transverse isotropy for orthopedic porous implants

Additively manufactured lattice structures enable the design of tissue scaffolds with tailored mechanical properties, which can be implemented in porous biomaterials. The adaptation of bone to physiological loads results in anisotropic bone tissue properties which are optimized for site-specific loads; therefore, some bone sites are stiffer and stronger along the principal load direction compared to other orientations. In this work, a semi-analytical model was developed for the design of transversely isotropic lattice structures that can mimic the anisotropy characteristics of different types of bone tissue. Several design possibilities were explored, and a particular unit cell, which was best suited for additive manufacturing was further analyzed. The design of the unit cell was parameterized and in-silico analysis was performed via Finite Element Analysis. The structures were manufactured additively in metal and tested under compressive loads in different orientations. Finite element analysis showed good correlation with the semi-analytical model, especially for elastic constants with low relative densities. The anisotropy measured experimentally showed a variable accuracy, highlighting the deviations from designs to additively manufactured parts. Overall, the proposed model enables to exploit the anisotropy of lattice structures to design lighter scaffolds with higher porosity and increased permeability by aligning the scaffold with the principal direction of the load.

Inverse Problem in Resonant Ultrasound Spectroscopy With Sampling and Optimization: A Comparative Study on Human Cortical Bone

The Bayesian inference with prior knowledge has been proposed recently to solve the inverse problem in resonant ultrasound spectroscopy. It allows inferring the elastic properties of high damping materials, such as cortical bone with less dependence on the initial guessed values. In this method, the estimation of the stiffness coefficients is expressed as a probabilistic solution to the inverse problem, which can be achieved by sampling or optimization methods. However, the detailed performance comparison of these two strategies applied to high damping materials has not been fully studied. In this work, the full stiffness tensor of 52 transversely isotropic cortical bone specimens was obtained using Markov chain Monte Carlo (MCMC) sampling and particle swarm optimization (PSO), respectively. Results showed that the local probability distributions of stiffness coefficients estimated by the two methods are consistent. Compared with MCMC, the average calculation speed of PSO is ten times faster [614 s ± 59 s (MCMC) versus 53 s ± 22 s (PSO)]. The mean standard error between theoretical and experimental resonant frequencies was slightly smaller for PSO compared with MCMC. In conclusion, PSO, a global optimization strategy, is suitable to solve the inverse problem for high damping materials.

Application of differential evolution on elasticity measurement of low quality factor materials using FEM-based resonant ultrasound spectroscopy

Finite element method based resonant ultrasound spectroscopy (FEM-based RUS) allows elasticity measurement for a material with high quality factor (Q) and arbitrary geometry by minimizing the differences between its theoretically calculated resonant frequencies and the corresponding experimentally measured ones. As Q decreases, some experimental frequencies remain undetermined, which makes it difficult to pair the calculated and experimental frequencies and to correctly identify the elastic constants. Additional difficulty need be tackled for irregularly-shaped low-Q materials due to the adoption of time-consuming FEM, thus efficiency of the identification method needs to be focused on. To apply FEM-based RUS to low-Q materials, a new elastic constant identification method is proposed based on a differential evolution algorithm in this paper. This method can perform a global search combining with local optimizations in the elastic constant space, and improve the overall efficiency by limiting the number of the frequency calculations. By using numerical experiments, the effectiveness of the proposed method under different frequency missing situations was verified and its efficiency was measured from the required frequency calculation numbers, showing an approximate two third reduction compared with an existing method. Finally, the elastic constants of an actual irregular cortical bone-mimicking material (Q ≈ 25) were measured using the two methods, yielding consistent Young's moduli (calculated from the identified constants) with the data provided by the manufacturer and a similar improvement in computational efficiency of the proposed method.

A new elastic characterization method for anisotropic bilayer specimens via Bayesian resonant ultrasound spectroscopy

A novel nondestructive method for complete elastic characterization of substrate-coating bilayer specimens with distinct anisotropic layers via resonant ultrasound spectroscopy (RUS) and Bayesian inversion is developed here. Bayesian formulations of the RUS inversion problem-of quantifying elastic properties given a measured list of resonance frequencies recorded from a single, typically small, precisely fabricated, macroscopically homogeneous, linear-elastic specimen-are a recent development. Here we report the first Bayesian formulation of the bilayer problem, and through a series of practical examples, demonstrate novel parameter estimation capabilities of our open-source CmdStan-RUS code. Finding specimen geometry and the number of resonance modes used for inversion strongly govern the ability to retrieve individual elastic moduli. The concept of "invertability" is explored for a range of relevant geometries using virtual specimens that resemble experimental bilayers of plasma sprayed ceramic coatings on single crystal metallic substrates. A range of Bayesian posterior evaluation methods are addressed, particularly considering the large computational cost of the bilayer forward model. Laplace approximation methods are thus developed and implemented for bilayer geometry design space modeling and expedient estimates of parameter uncertainties. Ideal specimen design, different noise models, the influence of prior distributions, dual-likelihood fits incorporating measurements of the bare substrate, and how Bayesian RUS methods differ from traditional RUS optimization are discussed.

Bulk Wave Velocities in Cortical Bone Reflect Porosity and Compression Strength

The goal of this study was to evaluate whether ultrasonic velocities in cortical bone can be considered as a proxy for mechanical quality of cortical bone tissue reflected by porosity and compression strength. Micro-computed tomography, compression mechanical testing and resonant ultrasound spectroscopy were used to assess, respectively, porosity, strength and velocity of bulk waves of both shear and longitudinal polarisations propagating along and perpendicular to osteons, in 92 cortical bone specimens from tibia and femur of elderly human donors. All velocities were significantly associated with strength (r = 0.65-0.83) and porosity (r = -0.64 to -0.77). Roughly, according to linear regression models, a decrease in velocity of 100 m/s corresponded to a loss of 20 MPa in strength (which is approximately 10% of the largest strength value) and to an increase in porosity of 5%. These results provide a rationale for the in vivo measurement of one or several velocities for the diagnosis of bone fragility.

The Use of a Vibro-Acoustic Based Method to Determine the Composite Material Properties of a Replicate Clavicle Bone Model

Replicate bones are widely used as an alternative for cadaveric bones for in vitro testing. These composite bone models are more easily available and show low inter-specimen variability compared to cadaveric bone models. The combination of in vitro testing with in silico models can provide further insights in the evaluation of the mechanical behavior of orthopedic implants. An accurate numerical representation of the experimental model is important to draw meaningful conclusions from the numerical predictions. This study aims to determine the elastic material constants of a commonly used composite clavicle model by combining acoustic experimental and numerical modal analysis. The difference between the experimental and finite element (FE) predicted natural frequencies was minimized by updating the elastic material constants of the transversely isotropic cortical bone analogue that are provided by the manufacturer. The longitudinal Young's modulus was reduced from 16.00 GPa to 12.88 GPa and the shear modulus was increased from 3.30 GPa to 4.53 GPa. These updated material properties resulted in an average natural frequency difference of 0.49% and a maximum difference of 1.73% between the FE predictions and the experimental results. The presented updated model aims to improve future research that focuses on mechanical simulations with clavicle composite bone models.

Ultrasounds could be considered as a future tool for probing growing bone properties

Juvenile bone growth is well described (physiological and anatomical) but there are still lacks of knowledge on intrinsic material properties. Our group has already published, on different samples, several studies on the assessment of intrinsic material properties of juvenile bone compared to material properties of adult bone. The purpose of this study was finally to combine different experimental modalities available (ultrasonic measurement, micro-Computed Tomography analysis, mechanical compression tests and biochemical measurements) applied on small cubic bone samples in order to gain insight into the multiparametric evaluation of bone quality. Differences were found between juvenile and adult groups in term of architectural parameters (Porosity Separation), Tissue Mineral Density (TMD), diagonal stiffness coefficients (C₃₃, C_44, C_55, C₆₆) and ratio between immature and mature cross-links (CX). Diagonal stiffness coefficients are more representative of the microstructural and biochemical parameters of child bone than of adult bone. We also found that compression modulus E was highly correlated with several microstructure parameters and CX in children group while it was not at all correlated in the adult group. Similar results were found for the CX which was linked to several microstructure parameters (TMD and E) only in the juvenile group. To our knowledge, this is the first time that, on a same sample, ultrasonic measurements have been combined with the assessment of mechanical and biochemical properties. It appears that ultrasonic measurements can provide relevant indicators of child bone quality (microstructural and biochemical parameters) which is promising for clinical application since, B-mode ultrasound is the preferred first-line modality over other more constraining imaging modalities (radiation, parent–child accessibility and access to the patient's bed) for pediatric patients.

Data-Driven Computational Simulation in Bone Mechanics

The data-driven approach was formally introduced in the field of computational mechanics just a few years ago, but it has gained increasing interest and application as disruptive technology in many other fields of physics and engineering. Although the fundamental bases of the method have been already settled, there are still many challenges to solve, which are often inherently linked to the problem at hand. In this paper, the data-driven methodology is applied to a particular problem in tissue biomechanics, a context where this approach is particularly suitable due to the difficulty in establishing accurate and general constitutive models, due to the intrinsic intra and inter-individual variability of the microstructure and associated mechanical properties of biological tissues. The problem addressed here corresponds to the characterization and mechanical simulation of a piece of cortical bone tissue. Cortical horse bone tissue was mechanically tested using a biaxial machine. The displacement field was obtained by means of digital image correlation and then transformed into strains by approximating the displacement derivatives in the bone virtual geometric image. These results, together with the approximated stress state, assumed as uniform in the small pieces tested, were used as input in the flowchart of the data-driven methodology to solve several numerical examples, which were compared with the corresponding classical model-based fitted solution. From these results, we conclude that the data-driven methodology is a useful tool to directly simulate problems of biomechanical interest without the imposition (model-free) of complex spatial and individually-varying constitutive laws. The presented data-driven approach recovers the natural spatial variation of the solution, resulting from the complex structure of bone tissue, i.e. heterogeneity, microstructural hierarchy and multifactorial architecture, making it possible to add the intrinsic stochasticity of biological tissues into the data set and into the numerical approach.

Bionic mechanical design and 3D printing of novel porous Ti6Al4V implants for biomedical applications

In maxillofacial surgery, there is a significant need for the design and fabrication of porous scaffolds with customizable bionic structures and mechanical properties suitable for bone tissue engineering. In this paper, we characterize the porous Ti6Al4V implant, which is one of the most promising and attractive biomedical applications due to the similarity of its modulus to human bones. We describe the mechanical properties of this implant, which we suggest is capable of providing important biological functions for bone tissue regeneration. We characterize a novel bionic design and fabrication process for porous implants. A design concept of "reducing dimensions and designing layer by layer" was used to construct layered slice and rod-connected mesh structure (LSRCMS) implants. Porous LSRCMS implants with different parameters and porosities were fabricated by selective laser melting (SLM). Printed samples were evaluated by microstructure characterization, specific mechanical properties were analyzed by mechanical tests, and finite element analysis was used to digitally calculate the stress characteristics of the LSRCMS under loading forces. Our results show that the samples fabricated by SLM had good structure printing quality with reasonable pore sizes. The porosity, pore size, and strut thickness of manufactured samples ranged from (60.95± 0.27)% to (81.23±0.32)%, (480±28) to (685±31) μm, and (263±28) to (265±28) μm, respectively. The compression results show that the Young's modulus and the yield strength ranged from (2.23±0.03) to (6.36±0.06) GPa and (21.36±0.42) to (122.85±3.85) MPa, respectively. We also show that the Young's modulus and yield strength of the LSRCMS samples can be predicted by the Gibson-Ashby model. Further, we prove the structural stability of our novel design by finite element analysis. Our results illustrate that our novel SLM-fabricated porous Ti6Al4V scaffolds based on an LSRCMS are a promising material for bone implants, and are potentially applicable to the field of bone defect repair.

Multiscale Characterisation of Cortical Bone Tissue

Multiscale analysis has become an attractive technique to predict the behaviour of materials whose microstructure strongly changes spatially or among samples, with that microstructure controlling the local constitutive behaviour. This is the case, for example, of most biological tissues—such as bone. Multiscale approaches not only allow, not only to better characterise the local behaviour, but also to predict the field-variable distributions (e.g., strains, stresses) at both scales (macro and micro) simultaneously. However, multiscale analysis usually lacks sufficient experimental feedback to demonstrate its validity. In this paper an experimental and numerical micromechanics analysis is developed with application to cortical bone. Displacement and strain fields are obtained across the microstructure by means of digital image correlation (DIC). The other mechanical variables are computed following the micromechanics theory. Special emphasis is given to the differences found in the different field variables between the micro- and macro-structures, which points out the need for this multiscale approach in cortical bone tissue. The obtained results are used to establish the basis of a multiscale methodology with application to the analysis of bone tissue mechanics at different spatial scales.

A resonant frequency retrieving method for low Q-factor materials based on resonant ultrasound spectroscopy

Resonant ultrasound spectroscopy (RUS) allows identification of the elastic properties of solid materials vibrating under an ultrasonic excitation from the measurement of their inherent frequencies. Retrieving the resonant frequencies is therefore a key signal processing step in RUS, which is generally addressed using a linear prediction filter. In this study, the Empirical Mode Decomposition (EMD) was proposed to retrieve the inherent resonant frequencies of materials with low Q-factor (quality factor). EMD was used to decompose the frequency response of the tested sample into intrinsic mode functions (IMF). The relevant IMF was selected from which the resonant frequencies could be computed. A bovine cortical bone sample was measured and its resonant frequencies were identified with EMD and with linear prediction for comparison. The elastic constants were also derived using both approaches. The number of resonant frequencies (45) extracted with EMD was larger than the number of frequencies (26) identified using the classical linear prediction approach. In particular, EMD proved to be more effective in detecting resonance in the higher frequency range (i.e., between 235 kHz and 400 kHz), i.e., on the weak excitation side where the spectral amplitude is low. The number of measured frequencies matching with the calculated ones was also larger for EMD (39) compared to linear prediction (17). If these results are confirmed in further studies on more samples, EMD combined with RUS, by improving the extraction of resonant frequencies for low Q-factor materials, may be considered to be useful not only to improve the reliability of the estimation of elastic parameters, but also to extend the application range of RUS.

Application of the Johnson-Cook plasticity model in the finite element simulations of the nanoindentation of the cortical bone

The mechanical behavior of the cortical bone in nanoindentation is a complicated mechanical problem. The finite element analysis has commonly been assumed to be the most appropriate approach to this issue. One significant problem in nanoindentation modeling of the elastic-plastic materials is pile-up deformation, which is not observed in cortical bone nanoindentation testing. This phenomenon depends on the work-hardening of materials; it doesn't occur for work-hardening materials, which suggests that the cortical bone could be considered as a work-hardening material. Furthermore, in a recent study [59], a plastic hardening until failure was observed on the micro-scale of a dry ovine osteonal bone samples subjected to micropillar compression. The purpose of the current study was to apply an isotropic hardening model in the finite element simulations of the nanoindentation of the cortical bone to predict its mechanical behavior. The Johnson-Cook (JC) model was chosen as the constitutive model. The finite element modeling in combination with numerical optimization was used to identify the unknown material constants and then the finite element solutions were compared to the experimental results. A good agreement of the numerical curves with the target loading curves was found and no pile-up was predicted. A Design Of Experiments (DOE) approach was performed to evaluate the linear effects of the material constants on the mechanical response of the material. The strain hardening modulus and the strain hardening exponent were the most influential parameters. While a positive effect was noticed with the Young's modulus, the initial yield stress and the strain hardening modulus, an opposite effect was found with the Poisson's ratio and the strain hardening exponent. Finally, the JC model showed a good capability to describe the elastoplastic behavior of the cortical bone.

Computed tomography porosity and spherical indentation for determining cortical bone millimetre-scale mechanical properties

The cortex of the femoral neck is a key structural element of the human body, yet there is not a reliable metric for predicting the mechanical properties of the bone in this critical region. This study explored the use of a range of non-destructive metrics to measure femoral neck cortical bone stiffness at the millimetre length scale. A range of testing methods and imaging techniques were assessed for their ability to measure or predict the mechanical properties of cortical bone samples obtained from the femoral neck of hip replacement patients. Techniques that can potentially be applied in vivo to measure bone stiffness, including computed tomography (CT), bulk wave ultrasound (BWUS) and indentation, were compared against in vitro techniques, including compression testing, density measurements and resonant ultrasound spectroscopy. Porosity, as measured by micro-CT, correlated with femoral neck cortical bone's elastic modulus and ultimate compressive strength at the millimetre length scale. Large-tip spherical indentation also correlated with bone mechanical properties at this length scale but to a lesser extent. As the elastic mechanical properties of cortical bone correlated with porosity, we would recommend further development of technologies that can safely measure cortical porosity in vivo.

Anisotropic elastic properties of human femoral cortical bone and relationships with composition and microstructure in elderly

The strong dependence between cortical bone elasticity at the millimetre-scale (mesoscale) and cortical porosity has been evidenced by previous studies. However, bone is an anisotropic composite material made by mineral, proteins and water assembled in a hierarchical structure. Whether the variations of structural and compositional properties of bone affect the different elastic coefficients at the mesoscale is not clear. Aiming to understand the relationships between bone elastic properties and compositions and microstructure, we applied state-of-the-art experimental modalities to assess these aspects of bone characteristics. All elastic coefficients (stiffness tensor of the transverse isotropic bone material), structure of the vascular pore network, collagen and mineral properties were measured in 52 specimens from the femoral diaphysis of 26 elderly donors. Statistical analyses and micromechanical modeling showed that vascular pore volume fraction and the degree of mineralization of bone are the most important determinants of cortical bone anisotropic mesoscopic elasticity. Though significant correlations were observed between collagen properties and elasticity, their effects in bone mesoscopic elasticity were minor in our data. This work also provides a unique set of data exhibiting a range of variations of compositional and microstructural cortical bone properties in the elderly and gives strong experimental evidence and basis for further development of biomechanical models for human cortical bone. STATEMENT OF SIGNIFICANCE: This study reports the relationships between microstructure, composition and the mesoscale anisotropic elastic properties of human femoral cortical bone in elderly. For the first time, we provide data covering the complete anisotropic elastic tensor, the microstructure of cortical vascular porosity, mineral and collagen characteristics obtained from the same or adjacent samples in each donor. The results revealed that cortical vascular porosity and degree of mineralization of bone are the most important determinants of bone anisotropic stiffness at the mesoscale. The presented data gives strong experimental evidence and basis for further development of biomechanical models for human cortical bone.

Elastic constants identification of irregular hard biological tissue materials using FEM-based resonant ultrasound spectroscopy

This paper aims to apply the resonant ultrasound spectroscopy technique (RUS) combined with micro computed tomography (μ-CT) and finite element method (FEM) to quantify the elastic constants of the irregular hard biological tissue material such as enamel. In this method, the resonant frequencies of an irregular shaped sample tested under stress-free boundary conditions are measured first. Then, micro-computed tomography (μ-CT) is used to acquire three-dimensional (3-D) geometry information of the sample, and the resonant frequencies are calculated with FEM. Thereby, an optimization procedure using the Levenberg-Marquardt algorithm updates the elastic constants in the FEM model until the output natural frequencies from the model fit the results from the RUS experiments. The proposed method has been tested first on a calibration material. To this purpose, titanium has been selected. The elastic constants of a rectangular parallelepiped shaped titanium sample obtained by the conventional RUS method and those of five irregular samples obtained by FEM-based RUS were in good agreement, displaying differences less than 2%. Once the method has been validated on titanium, it was applied to an enamel sample. The results show that the FEM-based RUS method can effectively identify the elastic constants of irregular titanium and enamel samples. This study expands the application range of RUS technology and provides a new method for the measurement of elastic properties of irregular hard biological tissue materials.

Homogenization of cortical bone reveals that the organization and shape of pores marginally affect elasticity

With ageing and various diseases, the vascular pore volume fraction (porosity) in cortical bone increases, and the morphology of the pore network is altered. Cortical bone elasticity is known to decrease with increasing porosity, but the effect of the microstructure is largely unknown, while it has been thoroughly studied for trabecular bone. Also, popular micromechanical models have disregarded several micro-architectural features, idealizing pores as cylinders aligned with the axis of the diaphysis. The aim of this paper is to quantify the relative effects on cortical bone anisotropic elasticity of porosity and other descriptors of the pore network micro-architecture associated with pore number, size and shape. The five stiffness constants of bone assumed to be a transversely isotropic material were measured with resonant ultrasound spectroscopy in 55 specimens from the femoral diaphysis of 29 donors. The pore network, imaged with synchrotron radiation X-ray micro-computed tomography, was used to derive the pore descriptors and to build a homogenization model using the fast Fourier transform (FFT) method. The model was calibrated using experimental elasticity. A detailed analysis of the computed effective elasticity revealed in particular that porosity explains most of the variations of the five stiffness constants and that the effects of other micro-architectural features are small compared to usual experimental errors. We also have evidence that modelling the pore network as an ensemble of cylinders yields biased elasticity values compared to predictions based on the real micro-architecture. The FFT homogenization method is shown to be particularly efficient to model cortical bone.

Optimization of Taylor spatial frame half-pins diameter for bone deformity correction: Application to femur

Using external fixtures for bone deformity correction takes advantages of less soft tissue injury, better bone alignment and enhances strain development for bone formation on cutting section, which cause shorter healing time. Among these fixtures, Taylor spatial frame is widely used and includes two rings and six adjustable struts developing 6 degrees of freedom, making them very flexible for this type of application. The current study describes a method to optimize Taylor spatial frame pin-sizes currently chosen from the surgeon's experiences. A three-dimensional model of femur was created from computed tomography images; segmentation of the medical images was made based on the Hounsfield unit (gray scale) in order to allocate adequate mechanical properties into cortical and trabecular bone sections. Both the cortical and trabecular sections were assumed to be isotropic and homogeneous. The diameter optimization of Taylor spatial frame's half-pins was carried out by coupling genetic algorithm and finite element analysis. The finite element analysis was based on a static mechanical load corresponding to a standing person's body weight. Finite element analysis results were validated with experimentally measured strains obtained from bone compression tests. A cost function, based on the developed bone stresses, was defined close to the Taylor spatial frame's half-pins. The calculated cost function showed a decrease of over 33% from the initial half-pin selection by the surgeon and the genetic algorithm optimization. Consequently, the maximum stresses experienced by the bone in the connected location of the half-pins decreased from 121.4 MPa in the surgeon's selection to 73.07 MPa as a result of the optimization process.

Assessment of elastic coefficients of child cortical bone using resonant ultrasound spectroscopy

The assessment of the anisotropic elastic properties of non-pathological child cortical bone remains a challenge for the biomechanical engineering community and an important clinical issue. Resonant ultrasound spectroscopy (RUS) can be used to determine bone stiffness coefficients from the mechanical resonances of bone specimens. Here, a RUS protocol was used on 7 fibula specimens from children (mean age 14 ± 3 years) to estimate the whole elastic stiffness tensor of non-pathological child cortical bone considered as orthotropic. Despite a small number of sample, results are consistent with this hypothesis, even if a trend towards transverse isotropy is discussed. Indeed, the average values of the 9 independent stiffness coefficients obtained in this study for child bone are: C₁₁ = 16.73 ± 0.19 GPa, C₂₂ = 16.19 ± 0.12 GPa, C₃₃ = 24.47 ± 0.30 GPa, C₄₄ = 4.14 ± 0.08 GPa, C₅₅ = 4.16 ± 0.07 GPa, C₆₆ = 3.13 ± 0.05 GPa, C₁₂ = 10.14 ± 0.20 GPa, C₁₃ = 10.67 ± 0.27 GPa, C₂₃ = 10.25 ± 0.14 GPa.

Elastic properties measurement of human enamel based on resonant ultrasound spectroscopy

OBJECTIVES

To investigate the elastic properties of human enamel using resonant ultrasound spectroscopy (RUS).

METHODS

Six rectangular parallelepiped specimens were prepared from six human third molars. For all specimens, the theoretical resonant frequencies were calculated using the Rayleigh-Ritz method, knowing the specimen mass density and dimensions, and using a priori stiffness constants. The experimental resonant frequencies were measured and extracted by RUS. Then, the optimal stiffness constants were retrieved by adjustment of the theoretical resonant frequencies to the measured ones based on the Levenberg-Marquardt method. The engineering elastic moduli, including Young's moduli, shear moduli, and Poisson's ratios, were also calculated based on the optimal stiffness constants.

RESULTS

The five independent stiffness constants C₁₁, C₁₂, C₁₃, C₃₃, and C₄₄ were 90.2 ± 6.65 GPa, 34.7 ± 6.90 GPa, 29.5 ± 4.82 GPa, 83.5 ± 8.93 GPa, and 37.0 ± 10.9 GPa, respectively. Young's moduli E₁₁ and E₃₃, shear moduli G₁₃ and G₁₂, and Poisson's ratios υ₁₂ and υ₁₃ were 71.7 ± 7.34 GPa, 69.2 ± 7.32 GPa, 37.0 ± 10.9 GPa, 28.1 ± 4.35 GPa, 0.303 ± 0.098, and 0.248 ± 0.060, respectively.

SIGNIFICANCE

Elastic properties are critical for developing dental materials and designing dental prostheses. The RUS method may provide more precise measurement of elastic properties of dental materials.

Measuring bone stiffness using spherical indentation

Objectives

Bone material properties are a major determinant of bone health in older age, both in terms of fracture risk and implant fixation, in orthopaedics and dentistry. Bone is an anisotropic and hierarchical material so its measured material properties depend upon the scale of metric used. The scale used should reflect the clinical problem, whether it is fracture risk, a whole bone problem, or implant stability, at the millimetre-scale. Indentation, an engineering technique involving pressing a hard-tipped material into another material with a known force, may be able to assess bone stiffness at the millimetre-scale (the apparent elastic modulus). We aimed to investigate whether spherical-tip indentation could reliably measure the apparent elastic modulus of human cortical bone.

Materials and methods

Cortical bone samples were retrieved from the femoral necks of nineteen patients undergoing total hip replacement surgery (10 females, 9 males, mean age: 69 years). The samples underwent indentation using a 1.5 mm diameter, ruby, spherical indenter tip, with sixty indentations per patient sample, across six locations on the bone surfaces, with ten repeated indentations at each of the six locations. The samples then underwent mechanical compression testing. The repeatability of indentation measurements of elastic modulus was assessed using the co-efficient of repeatability and the correlation between the bone elastic modulus measured by indentation and compression testing was analysed by least-squares regression.

Results

In total, 1140 indentations in total were performed. Indentation was found to be repeatable for indentations performed at the same locations on the bone samples with a mean co-efficient of repeatability of 0.4 GigaPascals (GPa), confidence interval (C.I): 0.33–0.42 GPa. There was variation in the indentation modulus results between different locations on the bone samples (mean co-efficient of repeatability: 3.1 GPa, C.I: 2.2–3.90 GPa). No clear correlation was observed between indentation and compression values of bone elastic modulus (r = 0.33, p = 0.17). The mean apparent elastic modulus obtained by spherical indentation was 9.9 GPa, the standard deviation for each indent cycle was 0.11 GPa, and the standard deviation between locations on the same sample was 1.01 GPa. The mean compression apparent elastic modulus was 4.42 GPa, standard deviation 1.02 GPa.

Discussion

Spherical-tip indentation was found to be a repeatable test for measuring the elastic modulus of human cortical bone, demonstrated by a low co-efficient of repeatability in this study. It could not, however, reliably predict cortical bone elastic modulus determined by platens compression testing in this study. This may be due to indentation only probing mechanical properties at the micro-scale while platens compression testing assesses millimetre length-scale properties. Improvements to the testing technique, including the use of a larger diameter spherical indenter tip, may improve the measurement of bone stiffness at the millimetre scale and should be investigated further.

Additive manufacturing technique-designed metallic porous implants for clinical application in orthopedics

Traditional metallic scaffold prostheses, as vastly applied implants in clinical orthopedic operations, have achieved great success in rebuilding limb function. However, mismatch of bone defects and additional coating requirements limit the long-term survival of traditional prostheses. Recently, additive manufacturing (AM) has opened up unprecedented possibilities for producing complicated structures in prosthesis shapes and microporous surface designs of customized prostheses, which can solve the drawback of traditional prostheses mentioned above. This review presents the most commonly used metallic additive manufacturing techniques, the microporous structure design of metallic scaffolds, and novel applications of customized prostheses in the orthopedic field. Challenges and future perspectives on AM fabricated scaffolds are also summarized.

Quantification of stiffness measurement errors in resonant ultrasound spectroscopy of human cortical bone

Resonant ultrasound spectroscopy (RUS) is the state-of-the-art method used to investigate the elastic properties of anisotropic solids. Recently, RUS was applied to measure human cortical bone, an anisotropic material with low Q-factor (20), which is challenging due to the difficulty in retrieving resonant frequencies. Determining the precision of the estimated stiffness constants is not straightforward because RUS is an indirect method involving minimizing the distance between measured and calculated resonant frequencies using a model. This work was motivated by the need to quantify the errors on stiffness constants due to different error sources in RUS, including uncertainties on the resonant frequencies and specimen dimensions and imperfect rectangular parallelepiped (RP) specimen geometry. The errors were first investigated using Monte Carlo simulations with typical uncertainty values of experimentally measured resonant frequencies and dimensions assuming a perfect RP geometry. Second, the exact specimen geometry of a set of bone specimens were recorded by synchrotron radiation micro-computed tomography. Then, a "virtual" RUS experiment is proposed to quantify the errors induced by imperfect geometry. Results show that for a bone specimen of ∼1° perpendicularity and parallelism errors, an accuracy of a few percent ( <6.2%) for all the stiffness constants and engineering moduli is achievable.

Mechanical characterization of structurally porous biomaterials built via additive manufacturing: experiments, predictive models, and design maps for load-bearing bone replacement implants

Porous biomaterials can be additively manufactured with micro-architecture tailored to satisfy the stringent mechano-biological requirements imposed by bone replacement implants. In a previous investigation, we introduced structurally porous biomaterials, featuring strength five times stronger than commercially available porous materials, and confirmed their bone ingrowth capability in an in vivo canine model. While encouraging, the manufactured biomaterials showed geometric mismatches between their internal porous architecture and that of its as-designed counterpart, as well as discrepancies between predicted and tested mechanical properties, issues not fully elucidated. In this work, we propose a systematic approach integrating computed tomography, mechanical testing, and statistical analysis of geometric imperfections to generate statistical based numerical models of high-strength additively manufactured porous biomaterials. The method is used to develop morphology and mechanical maps that illustrate the role played by pore size, porosity, strut thickness, and topology on the relations governing their elastic modulus and compressive yield strength. Overall, there are mismatches between the mechanical properties of ideal-geometry models and as-manufactured porous biomaterials with average errors of 49% and 41% respectively for compressive elastic modulus and yield strength. The proposed methodology gives more accurate predictions for the compressive stiffness and the compressive strength properties with a reduction of the average error to 11% and 7.6%. The implications of the results and the methodology here introduced are discussed in the relevant biomechanical and clinical context, with insight that highlights promises and limitations of additively manufactured porous biomaterials for load-bearing bone replacement implants.

STATEMENT OF SIGNIFICANCE

In this work, we perform mechanical characterization of load-bearing porous biomaterials for bone replacement over their entire design space. Results capture the shift in geometry and mechanical properties between as-designed and as-manufactured biomaterials induced by additive manufacturing. Characterization of this shift is crucial to ensure appropriate manufacturing of bone replacement implants that enable biological fixation through bone ingrowth as well as mechanical property harmonization with the native bone tissue. In addition, we propose a method to include manufacturing imperfections in the numerical models that can reduce the discrepancy between predicted and tested properties. The results give insight into the use of structurally porous biomaterials for the design and additive fabrication of load-bearing implants for bone replacement.

A critical assessment of the in-vitro measurement of cortical bone stiffness with ultrasound

Elasticity assessment based on bulk wave velocity (BWV) measurements is the most popular technique to characterize the anisotropic stiffness tensor in cortical bone. Typically, a cuboid bone specimen is cut with its sides along the different anatomical directions. Then, the velocity of shear and longitudinal waves propagating along different directions are assessed, from which stiffness coefficients are calculated. Despite the importance of obtaining accurate elasticity values for bone research, there is no generally accepted protocol to measure BWV and the precision of the technique has been seldom investigated. The purpose of this work is to critically assess the method to measure BWV on cuboid specimens in terms of ultrasound frequency, specimen size and signal processing technique. In this study, we measured polycarbonate specimens of different dimensions and 55 human bone specimens with different transducers using frequencies ranging from 2.25 to 10MHz and 1-5MHz for longitudinal and shear waves, respectively. We compared four signal processing methods to detect the wave arrival time. The main results are that, (1) the measurement of shear waves is more complex than that of longitudinal wave, being less precise and more sensitive to sample size; (2) the estimated stiffness depends on the signal processing technique used (up to 10% variation for shear coefficients of bone); and (3) bone stiffness assessed from BWV using the first arrival of the signal to determine the time-of-flight is not different from stiffness assessed using resonant ultrasound spectroscopy (RUS). These results evidence that the measurement method can have an effect on the stiffness values estimates and hence, a well-defined protocol is needed to accurately measure bone stiffness coefficients based on BWV.

The micro-damage process zone during transverse cortical bone fracture: No ears at crack growth initiation

OBJECTIVE

Apply high-resolution benchtop micro-computed tomography (micro-CT) to gain greater understanding and knowledge of the formation of the micro-damage process zone formed during traverse fracture of cortical bone.

METHODS

Bovine cortical bone was cut into single edge notch (bending) fracture testing specimens with the crack on the transverse plane and oriented to grow in the circumferential direction. We used a multi-specimen technique and deformed the specimens to various individual secant modulus loss levels (P-values) up to and including maximum load (Pmax). Next, the specimens were infiltrated with a BaSO₄ precipitation stain and scanned at 3.57-μm isotropic voxel size using a benchtop high resolution-micro-CT. Measurements of the micro-damage process zone volume, width and height were made. These were compared with the simple Irwin's process zone model and with finite element models. Electron and confocal microscopy confirmed the formation of BaSO₄ precipitate in micro-cracks and other porosity, and an interesting novel mechanism similar to tunneling.

RESULTS

Measurable micro-damage was detected at low P values and the volume of the process zone increased according to a second order polynomial trend. Both width and height grew linearly up to Pmax, at which point the process zone cross-section (perpendicular to the plane of the crack) was almost circular on average with a radius of approximately 550µm (approximately one quarter of the unbroken ligament thickness) and corresponding to the shape expected for a biological composite under plane stress conditions.

CONCLUSION

This study reports details of the micro-damage fracture process zone previously unreported for cortical bone. High-resolution micro-CT enables 3D visualization and measurement of the process zone and confirmation that the crack front edge and process zone are affected by microstructure. It is clear that the process zone for the specimens studied grows to be meaningfully large, confirming the need for the J-integral approach and it does not achieve steady state at Pmax in most specimens. With further development, this approach may become valuable towards better understanding the role of the process zone in cortical bone fracture and the effects of relevant modifications towards changes in fracture toughness in a cost effective way.

Measurement of human enamel mechanical characteristics with resonant ultrasound spectroscopy

The measurement of enamel mechanical properties offers great significance for the design, development and evaluation of clinical dental restorative materials. In this paper, resonant ultrasound spectroscopy (RUS) was proposed to measure the full second-order elastic tensor of human enamel specimens, and the mechanical parameters of human enamel, including Young's moduli, shear moduli and Poisson's ratios, were further calculated, which ranged from 64.50 to 80.46 GPa, 26.63 to 51.86 GPa and 0.18 to 0.40, separately. This study demonstrates that RUS shows feasibility on measuring the mechanical properties of human enamel with repeatable and nondestructive advantages.

The Elasticity Coefficients Measurement of Human Dentin Based on RUS

This paper proposed to take advantages of resonant ultrasound spectroscopy (RUS) to measure the mechanical properties of human dentin specimen. The resonant spectroscopy of the dentin specimen was obtained between the frequency bands 155 and 575 kHz, and resonant frequencies were extracted by linear predictive filter and then by Levenberg-Marquardt method. By inverse problem approach, 13 experimental resonant frequencies progressively matched to the first 30 orders of theoretical resonant frequencies calculated by Lagrangian variational method. The full second-order elastic tensor of dentin specimen was adjusted. The whole set of human dentin engineering moduli, including Young's moduli (E₁₁ = 22.641 GPa, E₃₃ = 13.637 GPa), shear moduli (G₁₂ = 10.608 GPa, G₂₃ = 7.742 Gpa), and Poisson's ratios (ν₁₂ = 0.067, ν₃₁ = 0.378), were finally calculated. This study demonstrates that RUS can be successfully adapted to measure the mechanical properties of low quality factor biomaterials.

Development of a methacrylate-terminated PLGA copolymer for potential use in craniomaxillofacial fracture plates

We synthesised methacrylate-terminated PLGA (HT-PLGA, 85:15 LA:GA, 169kDa), for potential use as an adhesively attached craniomaxillofacial fracture fixation plate. The in vitro degradation of molecular weight, pH and flexural modulus were measured over 6weeks storage in PBS at 37°C, with commercially available high (225kDa, H-PLGA) and low (116kDa, L-PLGA) molecular weight 85:15 PLGAs used as comparators. Molecular weights of the materials reduced over 6weeks, HT-PLGA by 48%, H-PLGA by 23% and L-PLGA by 81%. HT-PLGA and H-PLGA exhibited a near constant pH (7.35) and had average flexural moduli in excess of 6GPa when produced, similar to that of the mandible. After 1week storage both exhibited a significant reduction in average modulus, however, from weeks 1-6 no further significant changes were observed, the average modulus never dropped significantly below 5.5GPa. In contrast, the L-PLGA caused a pH drop to below 7.3 by week 6 and an average modulus drop to 0.6 from an initial 4.6GPa. Cell culture using rat bone marrow stromal cells, revealed all materials were cytocompatible and exhibited no osteogenic potential. We conclude that our functionalised PLGA retains mechanical properties which are suitable for use in craniofacial fixation plates.

Topological design and additive manufacturing of porous metals for bone scaffolds and orthopaedic implants: A review

One of the critical issues in orthopaedic regenerative medicine is the design of bone scaffolds and implants that replicate the biomechanical properties of the host bones. Porous metals have found themselves to be suitable candidates for repairing or replacing the damaged bones since their stiffness and porosity can be adjusted on demands. Another advantage of porous metals lies in their open space for the in-growth of bone tissue, hence accelerating the osseointegration process. The fabrication of porous metals has been extensively explored over decades, however only limited controls over the internal architecture can be achieved by the conventional processes. Recent advances in additive manufacturing have provided unprecedented opportunities for producing complex structures to meet the increasing demands for implants with customized mechanical performance. At the same time, topology optimization techniques have been developed to enable the internal architecture of porous metals to be designed to achieve specified mechanical properties at will. Thus implants designed via the topology optimization approach and produced by additive manufacturing are of great interest. This paper reviews the state-of-the-art of topological design and manufacturing processes of various types of porous metals, in particular for titanium alloys, biodegradable metals and shape memory alloys. This review also identifies the limitations of current techniques and addresses the directions for future investigations.

Distribution of mesoscale elastic properties and mass density in the human femoral shaft

Cortical bone properties are determined by tissue composition and structure at several hierarchical length scales. In this study, the spatial distribution of micro- and mesoscale elastic properties within a human femoral shaft has been investigated. Microscale tissue degree of mineralization (DMB), cortical vascular porosity Ct.Po and the average transverse isotropic stiffness tensor C(Micro) of cylindrical-shaped samples (diameter: 4.4 mm, N = 56) were obtained from cortical regions between 20 and 85% of the total femur length and around the periphery (anterior, medial, posterior and lateral quadrants) by means of synchrotron radiation µCT (SRµCT) and 50-MHz scanning acoustic microscopy (SAM). Within each cylinder, the volumetric bone mineral density (vBMD) and the mesoscale stiffness tensor C(Meso) were derived using a numerical homogenization approach. Moreover, microelastic maps of the axial elastic coefficient c33 measured by SAM at distinct cross-sectional locations along the femur were used to construct a 3-D multiscale elastic model of the femoral shaft. Variations of vBMD (6.1%) were much lower than the variations of mesoscale elastic coefficients (11.1-21.3%). The variation of DMB was only a minor predictor for variations of the mesoscale elastic properties (0.05 ≤ R(2) ≤ 0.34). Instead, variations of the mesoscale elastic properties could be explained by variations of cortical porosity and microscale elastic properties. These data were suitable inputs for numerical evaluations and may help to unravel the relations between structure and composition on the elastic function in cortical bone.

To what extent can cortical bone millimeter-scale elasticity be predicted by a two-phase composite model with variable porosity?

An evidence gap exists in fully understanding and reliably modeling the variations in elastic anisotropy that are observed at the millimeter scale in human cortical bone. The porosity (pore volume fraction) is known to account for a large part, but not all, of the elasticity variations. This effect may be modeled by a two-phase micromechanical model consisting of a homogeneous matrix pervaded by cylindrical pores. Although this model has been widely used, it lacks experimental validation. The aim of the present work is to revisit experimental data (elastic coefficients, porosity) previously obtained from 21 cortical bone specimens from the femoral mid-diaphysis of 10 donors and test the validity of the model by proposing a detailed discussion of its hypotheses. This includes investigating to what extent the experimental uncertainties, pore network modeling, and matrix elastic properties influence the model's predictions. The results support the validity of the two-phase model of cortical bone which assumes that the essential source of variations of elastic properties at the millimeter-scale is the volume fraction of vascular porosity. We propose that the bulk of the remaining discrepancies between predicted stiffness coefficients and experimental data (RMSE between 6% and 9%) is in part due to experimental errors and part due to small variations of the extravascular matrix properties. More significantly, although most of the models that have been proposed for cortical bone were based on several homogenization steps and a large number of variable parameters, we show that a model with a single parameter, namely the volume fraction of vascular porosity, is a suitable representation for cortical bone. The results could provide a guide to build specimen-specific cortical bone models. This will be of interest to analyze the structure-function relationship in bone and to design bone-mimicking materials.

Combined estimation of thickness and velocities using ultrasound guided waves: a pioneering study on in vitro cortical bone samples

This paper reports for the first time on inverse estimation of several bone properties from guided-wave measurements in human bone samples. Previously, related approaches have focused on ultrasonic estimation of a single bone property at a time. The method is based on two steps: the multi-Lamb mode response is analyzed using the singular value decomposition signal processing method recently introduced in the field, then an identification procedure is run to find thickness and anisotropic elastic properties of the considered specimen. Prior to the measurements on bone, the method is validated on cortical bone-mimicking phantoms. The repeatability and the trueness of the estimated parameters on bone-mimicking phantoms were found around a few percent. Estimation of cortical thickness on bone samples was in good agreement with cortical thickness derived from high-resolution peripheral quantitative computed tomography data analysis of the samples.