Estimation of the in-plane ultimate stress of lamellar tissue as a function of bone mineral density and osteocyte lacunae porosity.

BACKGROUND AND OBJECTIVE: Detailed finite element models based on medical images (µ-CT) are commonly used to analyze the mechanical behavior of bone at microscale. In order to simulate the tissue failure onset, isotropic failure criteria of lamellar tissue are often used, despite its non-isotropic and heterogeneous nature. The main goal of the present work is to estimate the in-plane ultimate stress of lamellar bone, considering the influence of mineral content and the porosity due to the osteocyte lacunae concentration. METHODS: To this aim, a representative volume cell of lamellar tissue is modeled numerically, including: (1) non-isotropic elastic properties of tissue as a function of the bone mineral density and (2) explicit modeling of the osteocyte lacunae, considering the range of porosity content, size and orientation of ellipsoid-shaped lacunae. Firstly, the element size for the finite element models have been defined from a preliminary convergence analysis. Bounds on the ultimate stress of non-porous lamellar tissue are estimated for two values of bone mineral density, considering the results of tensile and compressive tests of wet osteons from the literature. Subsequently, the ultimate stress of lamellar tissue considering several values of micro-porosity is addressed. RESULTS: Results obtained in this work show that the strength of lamellar bone decreases exponentially with the increase of lacunae porosity concentration. Ultimate stress of non-porous tissue (p=0%) increases with high mineral content, reaching a value of S¯transc=355.40±39.80 MPa for compression in the transversal direction of the fiber bundles, being BMD=1.246g/cm3. The mean value for the longitudinal to transverse strength ratio evaluated for porosity p=0%,1% and 5% and a mineral content BMD=1.2g/cm3, is 2.47:1 for tension and 1.55:1 for compression. These values are in agreement with literature. CONCLUSIONS: Osteocyte lacunae act as stress concentrators, acting as potential stimulus for the bone regeneration process. A novel micromechanical model for the in-plane ultimate stress of lamellar tissue as a function of mineral content and lacunae concentration is presented. Additional considerations about the intralamellar shear stress evolution are also given.

Numerical analysis of the influence of triply periodic minimal surface structures morphometry on the mechanical response.

BACKGROUND AND OBJECTIVE: Design of bone scaffolds requires a combination of material and geometry to fulfil requirements of mechanical properties, porosity and pore size. Triply Periodic Minimal Surface (TPMS) structures have gained attention due to their similarities to cancellous bone. In this work, we aim at exploring relationships between morphometry and mechanical properties for TPMS configurations. METHODS: Eight TPMS structures are defined considering six porosity levels and their morphometry is characterized. The stiffness matrix of each structure is assessed and related to morphometry through a statistical analysis. RESULTS: An orthotropic mechanical behavior has been derived from the numerical homogenization. Properties decay exponentially for decreasing volume fraction. Through volume fraction variation, TPMS mechanical properties can be selected to match bone properties in a range of 0.2% to 70% of the bulk material properties. CONCLUSIONS: The comparison between cancellous bone and TPMS morphometry, considering a unit cell size of 1.5 mm, reveals that the configurations analyzed in this work match the requirements of volume fraction, mean thickness and pore size. However, the TPMS studied in this work differ from cancellous bone anisotropy. The results in this paper provide a framework to select the proper TPMS configuration and its geometry for patient-specific applications.

Numerical modelling of cancellous bone damage using an orthotropic failure criterion and tissue elastic properties as a function of the mineral content and microporosity.

BACKGROUND AND OBJECTIVE: Elastic and strength properties of lamellar tissue are essential to analyze the mechanical behaviour of bone at the meso- or macro-scale. Although many efforts have been made to model the architecture of cancellous bone, in general, isotropic elastic constants are assumed for tissue modelling, neglecting its non-isotropic behaviour. Therefore, isotropic damage laws are often used to estimate the bone failure. The main goals of this work are: (1) to present a new model for the estimation of the elastic properties of lamellar tissue which includes the bone mineral density (BMD) and the microporosity, (2) to address the numerical modelling of cancellous bone damage using an orthotropic failure criterion and a discrete damage mechanics analysis, including the novel approach for the tissue elastic properties aforementioned. METHODS: Numerical homogenization has been used to estimate the elastic properties of lamellar bone considering BMD and microporosity. Microcomputed Tomography (µ-CT) scans have been performed to obtain the micro-finite element (µ-FE) model of cancellous bone from a vertebra of swine. In this model, lamellar tissue is orientated by considering a unidirectional layer pattern being the mineralized collagen fibrils aligned with the most representative geometrical feature of the trabeculae network. We have considered the Hashin's failure criterion and the Material Property Degradation (MPDG) method for simulating the onset and evolution of bone damage. RESULTS: The terms of the stiffness matrix for lamellar tissue are derived as functions of the BMD and microporosity at tissue scale. Results obtained for the apparent yield strain values agree with experimental values found in the literature. The influence of the damage parameters on the bone mechanics behaviour is also presented. CONCLUSIONS: Stiffness matrix of lamellar tissue depends on both BMD and microporosity. The new approach presented in this work enables to analyze the influence of the BMD and porosity on the mechanical response of bone. Lamellar tissue orientation has to be considered in the mechanical analysis of the cancellous bone. An orthotropic failure criterion can be used to analyze the bone failure onset instead of isotropic criteria. The elastic property degradation method is an efficient procedure to analyze the failure propagation in a 3D numerical model.

Open cell polyurethane foam compression failure characterization and its relationship to morphometry.

Open cell polyurethane foams are often used as cancellous bone surrogates because of their similarities in morphology and mechanical response. In this work, open cell polyurethane foams of three different densities are characterized from morphometric and mechanical perspectives. The analysis of micro-computed tomography images has revealed that the high density foams present the greatest inhomogeneities. Those inhomogeneities promoted the failure location. We have used the finite element models as a tool to estimate elastic and failure properties that can be used in numerical modeling. Furthermore, we have assessed the anisotropic mechanical response of the foams, whose differences are related to the morphometric inhomogeneities. We found significant relationships between morphometry and the elastic and failure response. The detailed information about morphometry, elastic constants and strength limits provided in this work can be of interest to researchers and practitioners that often use these polyurethane foams in orthopedic implants and cement augmentation evaluations.

Some Practical Considerations for Compression Failure Characterization of Open-Cell Polyurethane Foams Using Digital Image Correlation.

(1) Background: Open-cell polyurethane foam mechanical behavior is highly influenced by microstructure. The determination of the failure mechanisms and the characterization of the deformation modes involved at the micro scale is relevant for accurate failure modeling. (2) Methods: We use digital image correlation (DIC) to investigate strain fields of open-cell polyurethane foams of three different densities submitted to compression testing. We analyze the effect of some DIC parameters on the failure pattern definition and the equivalent strain magnification at the apparent ultimate point. Moreover, we explore speckle versus non-speckle approaches and discuss the importance of determining the pattern quality to perform the displacement correlation. (3) Results: DIC accurately characterizes the failure patterns. A variation in the subset size has a relevant effect on the strain magnification values. Step size effect magnitude depends on the subset size. The pattern matching criterion presented little influence (3.5%) on the strain magnification. (4) Conclusion: The current work provides a comprehensive analysis of the influence of some DIC parameters on compression failure characterization of foamed structures. It highlights that changes of subset and step sizes have a significant effect on the failure pattern definition and the strain magnification values, while the pattern matching criterion and the use of speckle have a minor influence on the results. Moreover, this work stands out that the determination of the pattern quality has a major importance for texture analysis. The in-depth, detailed study carried out with samples of three different apparent densities is a useful guide for DIC users as regards texture correlation and foamed structures.

Explicit expressions for the estimation of the elastic constants of lamellar bone as a function of the volumetric mineral content using a multi-scale approach.

In this work, explicit expressions to estimate all the transversely isotropic elastic constants of lamellar bone as a function of the volumetric bone mineral density (BMD) are provided. The methodology presented is based on the direct homogenization procedure using the finite element method, the continuum approach based on the Hill bounds, the least-square method and the mean field technique. Firstly, a detailed description of the volumetric content of the different components of bone is provided. The parameters defined in this step are related to the volumetric BMD considering that bone mineralization process occurs at the smallest scale length of the bone tissue. Then, a thorough description provides the details of the numerical models and the assumptions adopted to estimate the elastic behaviour of the forward scale lengths. The results highlight the noticeable influence of the BMD on the elastic modulus of lamellar bone. Power law regressions fit the Young's moduli, shear stiffness moduli and Poisson ratios. In addition, the explicit expressions obtained are applied to the estimation of the elastic constants of cortical bone. At this scale length, a representative unit cell of cortical bone is analysed including the fibril orientation pattern given by Wagermaier et al. (Biointerphases 1:1-5, 2006) and the BMD distributions observed by Granke et al. (PLoS One 8:e58043, 2012) for the osteon. Results confirm that fibril orientation arrangement governs the anisotropic behaviour of cortical bone instead of the BMD distribution. The novel explicit expressions obtained in this work can be used for improving the accuracy of bone fracture risk assessment.

A multiscale modelling of bone ultrastructure elastic proprieties using finite elements simulation and neural network method.

Bone is a living material with a complex hierarchical structure which entails exceptional mechanical properties, including high fracture toughness, specific stiffness and strength. Bone tissue is essentially composed by two phases distributed in approximately 30-70%: an organic phase (mainly type I collagen and cells) and an inorganic phase (hydroxyapatite-HA-and water). The nanostructure of bone can be represented throughout three scale levels where different repetitive structural units or building blocks are found: at the first level, collagen molecules are arranged in a pentameric structure where mineral crystals grow in specific sites. This primary bone structure constitutes the mineralized collagen microfibril. A structural organization of inter-digitating microfibrils forms the mineralized collagen fibril which represents the second scale level. The third scale level corresponds to the mineralized collagen fibre which is composed by the binding of fibrils. The hierarchical nature of the bone tissue is largely responsible of their significant mechanical properties; consequently, this is a current outstanding research topic. Scarce works in literature correlates the elastic properties in the three scale levels at the bone nanoscale. The main goal of this work is to estimate the elastic properties of the bone tissue in a multiscale approach including a sensitivity analysis of the elastic behaviour at each length scale. This proposal is achieved by means of a novel hybrid multiscale modelling that involves neural network (NN) computations and finite elements method (FEM) analysis. The elastic properties are estimated using a neural network simulation that previously has been trained with the database results of the finite element models. In the results of this work, parametric analysis and averaged elastic constants for each length scale are provided. Likewise, the influence of the elastic constants of the tissue constituents is also depicted. Results highlight that intelligent numerical methods are powerful and accurate procedures to deal with the complex multiscale problem in the bone tissue with results in agreement with values found in literature for specific scale levels.

Influence of the mineral staggering on the elastic properties of the mineralized collagen fibril in lamellar bone.

In this work, a three-dimensional finite element model of the staggered distribution of the mineral within the mineralized collagen fibril has been developed to characterize the lamellar bone elastic behavior at the sub-micro length scale. Minerals have been assumed to be embedded in a collagen matrix, and different degrees of mineralization have been considered allowing the growth of platelet-shaped minerals both in the axial and the transverse directions of the fibril, through the variation of the lateral space between platelets. We provide numerical values and trends for all the elastic constants of the mineralized collagen fibril as a function of the volume fraction of mineral. In our results, we verify the high influence of the mineral overlapping on the mechanical response of the fibril and we highlight that the lateral distance between crystals is relevant to the mechanical behavior of the fibril and not only the mineral overlapping in the axial direction.