Development of bone surrogates by material extrusion-based additive manufacturing to mimic flexural mechanical behaviour and fracture prediction via phase-field approach.

BACKGROUND AND OBJECTIVE: The limited availability of human bone samples for investigation leads to the demand for alternatives. Bone surrogates are crucial in promoting research on the intricate mechanics of osseous tissue. However, solutions are restricted to commercial brands, which frequently fail to faithfully replicate the mechanical response of bone, or oversimplified customised simulants designed for a specific application. The manufacturing and assessment of reliable bone surrogates made of polylactic acid via material extrusion-based additive manufacturing are presented in this work. METHODS: An experimental and numerical study with 3D-printed dog-bone and prismatic specimens was carried out to characterise the polymeric feedstock and analyse the influence of process parameters under three-point bending and quasi-static conditions. Besides, three porcine rib samples were considered as a reference for the development of the artificial bones. Bone surrogates were manufactured from the 3D-scanned real bone geometries. In order to reproduce the trabecular and cortical bone, a lattice structure for the infill and a compact shell surrounding the core were employed. Infill density and shell thickness were evaluated through different printing configurations. Additionally, a computational analysis based on the phase-field approach was conducted to simulate the experimental tests and predict fracture. The modelling considered homogenisation of the infill material. RESULTS: Outcomes demonstrated the potential of the presented methodology. Maximum force and flexural stiffness were compared to real bone properties to find the optimal printing configuration, replicating the flexural mechanical behaviour of bone tissue. Certain configurations accurately reproduce the studied properties. Regarding the numerical model, strength and stiffness prediction was validated with experimental results. CONCLUSIONS: The presented methodology enables the manufacturing of artificial bones with accurate geometries and tailored mechanical properties. Furthermore, the described modelling strategy offers a powerful tool for designing bone surrogates.

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.

An Experimental and Numerical Investigation to Characterize an Aerospace Composite Material with Open-Hole Using Non-Destructive Techniques.

In this study, the open-hole quasi-static tensile and fatigue loading behavior of a multidirectional CFRP thick laminate, representative of laminates used in the aerospace industry, is studied. Non-destructive techniques such as infrared thermographic (IRT) and digital image correlation (DIC) are used to analyze the behavior of this material. We aim at characterizing the influence of the manufacturing defects and the stress concentrator through the temperature variation and strain distribution during fatigue and quasi-static tests. On the one hand, the fatigue specimens were tested in two main perpendicular directions of the laminate. The results revealed that manufacturing defects such as fiber waviness can have a major impact than open-hole stress concentrator on raising the material temperature and causing fracture. In addition, the number of plies with fibers oriented in the load direction can drastically reduce the temperature increment in the laminate. On the other hand, the quasi-static tensile tests showed that the strain distribution around the hole is able to predict the crack initiation and progression in the external plies. Finally, the experimental quasi-static tests were numerically simulated using the finite element method showing good agreement between the numerical and experimental results.

On the Role of Contact and System Stiffness in the Measurement of Principal Variables in Fretting Wear Testing.

In this work, the role of the contact stiffness in the measurement of principal variables in fretting wear tests is assessed. Several fretting wear tribometers found in the literature, including one developed by the authors, are analysed and modelled using numerical methods. The results show the importance of the tribosystem stiffness and tangential contact stiffness in the displacement sensor calibration and in the correct numerical modelling of fretting wear tests, especially for flat-to-flat contact configuration. The study highlights that, in most cases, direct comparisons between fretting results with severe wear obtained with different tribometers cannot be performed if the contact stiffness is not properly considered during the development of the experiments.

Numerical modelling of hip fracture patterns in human femur.

BACKGROUND AND OBJECTIVE: Hip fracture morphology is an important factor determining the ulterior surgical repair and treatment, because of the dependence of the treatment on fracture morphology. Although numerical modelling can be a valuable tool for fracture prediction, the simulation of femur fracture is not simple due to the complexity of bone architecture and the numerical techniques required for simulation of crack propagation. Numerical models assuming homogeneous fracture mechanical properties commonly fail in the prediction of fracture patterns. This paper focuses on the prediction of femur fracture based on the development of a finite element model able to simulate the generation of long crack paths. METHODS: The finite element model developed in this work demonstrates the capability of predicting fracture patterns under stance loading configuration, allowing the distinction between the main fracture paths: intracapsular and extracapsular fractures. It is worth noting the prediction of different fracture patterns for the same loading conditions, as observed during experimental tests. RESULTS AND CONCLUSIONS: The internal distribution of bone mineral density and femur geometry strongly influences the femur fracture morphology and fracture load. Experimental fracture paths have been analysed by means of micro-computed tomography allowing the comparison of predicted and experimental crack surfaces, confirming the good accuracy of the numerical model.

Assessment of mechanical properties of human head tissues for trauma modelling.

Many discrepancies are found in the literature regarding the damage and constitutive models for head tissues as well as the values of the constants involved in the constitutive equations. Their proper definition is required for consistent numerical model performance when predicting human head behaviour, and hence skull fracture and brain damage. The objective of this research is to perform a critical review of constitutive models and damage indicators describing human head tissue response under impact loading. A 3D finite element human head model has been generated by using computed tomography images, which has been validated through the comparison to experimental data in the literature. The threshold values of the skull and the scalp that lead to fracture have been analysed. We conclude that (1) compact bone properties are critical in skull fracture, (2) the elastic constants of the cerebrospinal fluid affect the intracranial pressure distribution, and (3) the consideration of brain tissue as a nearly incompressible solid with a high (but not complete) water content offers pressure responses consistent with the experimental data.

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.

Numerical Modelling of Femur Fracture and Experimental Validation Using Bone Simulant.

Bone fracture pattern prediction is still a challenge and an active field of research. The main goal of this article is to present a combined methodology (experimental and numerical) for femur fracture onset analysis. Experimental work includes the characterization of the mechanical properties and fracture testing on a bone simulant. The numerical work focuses on the development of a model whose material properties are provided by the characterization tests. The fracture location and the early stages of the crack propagation are modelled using the extended finite element method and the model is validated by fracture tests developed in the experimental work. It is shown that the accuracy of the numerical results strongly depends on a proper bone behaviour characterization.

A review on recent advances in numerical modelling of bone cutting.

Common practice of surgical treatments in orthopaedics and traumatology involves cutting processes of bone. These operations introduce risk of thermo-mechanical damage, since the threshold of critical temperature producing thermal osteonecrosis is very low. Therefore, it is important to develop predictive tools capable of simulating accurately the increase of temperature during bone cutting, being the modelling of these processes still a challenge. In addition, the prediction of cutting forces and mechanical damage is also important during machining operations. As the accuracy of simulations depends greatly on the proper choice of the thermo-mechanical properties, an essential part of the numerical model is the constitutive behaviour of the bone tissue, which is considered in different ways in the literature. This paper focuses on the review of the main contributions in modelling of bone cutting with special attention to the bone mechanical behaviour. The aim is to give the reader a complete vision of the approaches commonly presented in the literature in order to help in the development of accurate models for bone cutting.

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.

Numerical modelling of the mechanical behaviour of an osteon with microcracks.

In this work, we present two strategies for the numerical modelling of microcracks and damage within an osteon. A numerical model of a single osteon under compressive diametral load is developed, including lamellae organized concentrically around the haversian canal and the presence of lacunae. Elastic properties have been estimated from micromechanical models that consider the mineralized collagen fibrils reinforced with hydroxyapatite crystals and the dominating orientation of the fibrils in each lamella. Microcracks are simulated through the node release technique, enabling propagation along the lamellae interfaces by application of failure criteria initially conceived for composite materials, in particular the Brewer and Lagacé criterion for delamination. A second approach is also presented, which is based on the progressive degradation of the stiffness at the element level as the damage increases. Both strategies are discussed, showing a good agreement with experimental evidence reported by other authors. It is concluded that interlaminar shear stresses are the main cause of failure of an osteon under compressive diametral load.

Homogenized stiffness matrices for mineralized collagen fibrils and lamellar bone using unit cell finite element models.

Mineralized collagen fibrils have been usually analyzed like a two-phase composite material where crystals are considered as platelets that constitute the reinforcement phase. Different models have been used to describe the elastic behavior of the material. In this work, it is shown that when Halpin-Tsai equations are applied to estimate elastic constants from typical constituent properties, not all crystal dimensions yield a model that satisfy thermodynamic restrictions. We provide the ranges of platelet dimensions that lead to positive definite stiffness matrices. On the other hand, a finite element model of a mineralized collagen fibril unit cell under periodic boundary conditions is analyzed. By applying six canonical load cases, homogenized stiffness matrices are numerically calculated. Results show a monoclinic behavior of the mineralized collagen fibril. In addition, a 5-layer lamellar structure is also considered where crystals rotate in adjacent layers of a lamella. The stiffness matrix of each layer is calculated applying Lekhnitskii transformations, and a new finite element model under periodic boundary conditions is analyzed to calculate the homogenized 3D anisotropic stiffness matrix of a unit cell of lamellar bone. Results are compared with the rule-of-mixtures showing in general good agreement.