Layered water in crystal interfaces as source for bone viscoelasticity: arguments from a multiscale approach

The role of microenvironment in stem cell-based regeneration of intervertebral disc

Regenerative medicine for intervertebral disc (IVD) disease, by utilizing chondrocytes, IVD cells, and stem cells, has progressed to clinical trials in the treatment of back pain, and has been studied in various animal models of disc degeneration in the past decade. Stem cells exist in their natural microenvironment, which provides vital dynamic physical and chemical signals for their survival, proliferation and function. Long-term survival, function and fate of mesenchymal stem cells (MSCs) depend on the microenvironment in which they are transplanted. However, the transplanted MSCs and the endogenous disc cells were influenced by the complicated microenvironment in the degenerating disc with the changes of biochemical and biophysical components. It is important to understand how the MSCs and endogenous disc cells survive and thrive in the harsh microenvironment of the degenerative disc. Furthermore, materials containing stem cells and their natural microenvironment have good clinical effects. However, the implantation of tissue engineering IVD (TE-IVD) cannot provide a complete and dynamic microenvironment for MSCs. IVD graft substitutes may need further improvement to provide the best engineered MSC microenvironment. Additionally, the IVD progenitor cells inside the stem cell niches have been regarded as popular graft cells for IVD regeneration. However, it is still unclear whether actual IVD progenitor cells exist in degenerative spinal conditions. Therefore, the purpose of this review is fourfold: to discuss the presence of endogenous stem cells; to review and summarize the effects of the microenvironment in biological characteristics of MSC, especially those from IVD; to explore the feasibility and prospects of IVD graft substitutes and to elaborate state of the art in the use of MSC transplantation for IVD degeneration in vivo as well as their clinical application.

Bone hydration: How we can evaluate it, what can it tell us, and is it an effective therapeutic target?

Water constitutes roughly a quarter of the cortical bone by volume yet can greatly influence mechanical properties and tissue quality. There is a growing appreciation for how water can dynamically change due to age, disease, and treatment. A key emerging area related to bone mechanical and tissue properties lies in differentiating the role of water in its four different compartments, including free/pore water, water loosely bound at the collagen/mineral interfaces, water tightly bound within collagen triple helices, and structural water within the mineral. This review summarizes our current knowledge of bone water across the four functional compartments and discusses how alterations in each compartment relate to mechanical changes. It provides an overview on the advent of- and improvements to- imaging and spectroscopic techniques able to probe nano-and molecular scales of bone water. These technical advances have led to an emerging understanding of how bone water changes in various conditions, of which aging, chronic kidney disease, diabetes, osteoporosis, and osteogenesis imperfecta are reviewed. Finally, it summarizes work focused on therapeutically targeting water to improve mechanical properties.

Fiber Rearrangement and Matrix Compression in Soft Tissues: Multiscale Hypoelasticity and Application to Tendon

It is widely accepted that the nonlinear macroscopic mechanical behavior of soft tissue is governed by fiber straightening and re-orientation. Here, we provide a quantitative assessment of this phenomenon, by means of a continuum micromechanics approach. Given the negligibly small bending stiffness of crimped fibers, the latter are represented through a number of hypoelastic straight fiber phases with different orientations, being embedded into a hypoelastic matrix phase. The corresponding representative volume element (RVE) hosting these phases is subjected to “macroscopic” strain rates, which are downscaled to fiber and matrix strain rates on the one hand, and to fiber spins on the other hand. This gives quantitative access to the fiber decrimping (or straightening) phenomenon under non-affine conditions, i.e. in the case where the fiber orientations cannot be simply linked to the macroscopic strain state. In the case of tendinous tissue, such an RVE relates to the fascicle material with 50 μm characteristic length, made up of crimped collagen bundles and a gel-type matrix in-between. The fascicles themselves act as parallel fibers in a similar matrix at the scale of a tissue-related RVE with 500 μm characteristic length. As evidenced by a sensitivity analysis and confirmed by various mechanical tests, it is the initial crimping angle which drives both the degree of straightening and the shape of the macroscopic stress-strain curve, while the final linear portion of this curve depends almost exclusively on the collagen bundle elasticity. Our model also reveals the mechanical cooperation of the tissue’s key microstructural components: while the fibers carry tensile forces, the matrices undergo hydrostatic pressure.

Biological and bioinspired materials: Structure leading to functional and mechanical performance

Nature has achieved materials with properties and mechanisms that go far beyond the current know-how of the engineering-materials industry. The remarkable efficiency of biological materials, such as their exceptional properties that rely on weak constituents, high performance per unit mass, and diverse functionalities in addition to mechanical properties, has been mostly attributed to their hierarchical structure. Key strategies for bioinspired materials include formulating the fundamental understanding of biological materials that act as inspiration, correlating this fundamental understanding to engineering needs/problems, and fabricating hierarchically structured materials with enhanced properties accordingly. The vast, existing literature on biological and bioinspired materials can be discussed in terms of functional and mechanical aspects. Through essential representative properties and materials, the development of bioinspired materials utilizes the design strategies from biological systems to innovatively augment material performance for various practical applications, such as marine, aerospace, medical, and civil engineering. Despite the current challenges, bioinspired materials have become an important part in promoting innovations and breakthroughs in the modern materials industry.

Effect of geometrical structure variations on the viscoelastic and anisotropic behaviour of cortical bone using multi-scale finite element modelling

Multi-scale finite element analysis is performed to ascertain the effect of geometrical changes at multiple structural scales on the mechanical properties of cortical bone. Finite element models are developed, with reference to experimental data from existing literature, to account for bone's viscoelastic behaviour and anisotropic structure from the most fundamental level of bone consisting of mineralised collagen fibrils, up to the macroscopic level consisting of osteons and the Haversian canals. A statistical approach is incorporated to perform sensitivity analyses on the effects of different geometrical parameters on the effective material properties of cortical bone at each length scale. Numerical results indicate that there is an exponential correlation between the mineral volume fraction and the effective stiffness constants at each length scale. This contributes to the exponential behaviour of the instantaneous moduli describing cortical bone's two-phase stress relaxation process: a fast and slow response relaxation behaviour. Results indicate that the fast response relaxation time is independent of bone's structural anisotropy, whilst being dependent on variations in the global mineral volume fraction between length scales. However, the slow response relaxation time is independent of the changes in mineral volume fraction. It is also observed that the slow response relaxation time varies with bone's anisotropic structure, and therefore, contributes to the anisotropic properties of bone.

"Variances" and "in-variances" in hierarchical porosity and composition, across femoral tissues from cow, horse, ostrich, emu, pig, rabbit, and frog

It is very well known that bone is a hierarchically organized material produced by bone cells residing in the fluid environments filling (larger) vascular pores and (smaller) lacunar pores. The extracellular space consists of hydroxyapatite crystals, collagen type I molecules, and water with non-collageneous organics. It is less known to which extent the associated quantities (mineral, organic, and water concentrations; vascular, lacunar, and extracellular porosities) vary across species, organs, and ages. We here investigate the aforementioned quantities across femoral shaft tissues from cow, horse, emu, frog, ostrich, pig, and rabbit; by means of light microscopy and dehydration-demineralization tests; thereby revealing interesting invariances: The extracellular volume fractions of organic matter turn out to be similar across all tested non-amphibian tissues; as do the extracellular volume fractions of hydroxyapatite across all tested mammals. Hence, the chemical composition of the femoral extracellular bone matrix is remarkably "invariant" across differently aged mammals; while the water content shows significant variations, as does the partitions of water between the different pore spaces. The latter exhibit strikingly varying morphologies as well. This finding adds to the ample "universal patterns" in the sense of evolutionary developmental biology; and it provides interesting design requirements for the development of novel biomimetic tissue engineering solutions.

Patient-Specific Bone Multiscale Modelling, Fracture Simulation and Risk Analysis-A Survey

This paper provides a starting point for researchers and practitioners from biology, medicine, physics and engineering who can benefit from an up-to-date literature survey on patient-specific bone fracture modelling, simulation and risk analysis. This survey hints at a framework for devising realistic patient-specific bone fracture simulations. This paper has 18 sections: Section 1 presents the main interested parties; Section 2 explains the organzation of the text; Section 3 motivates further work on patient-specific bone fracture simulation; Section 4 motivates this survey; Section 5 concerns the collection of bibliographical references; Section 6 motivates the physico-mathematical approach to bone fracture; Section 7 presents the modelling of bone as a continuum; Section 8 categorizes the surveyed literature into a continuum mechanics framework; Section 9 concerns the computational modelling of bone geometry; Section 10 concerns the estimation of bone mechanical properties; Section 11 concerns the selection of boundary conditions representative of bone trauma; Section 12 concerns bone fracture simulation; Section 13 presents the multiscale structure of bone; Section 14 concerns the multiscale mathematical modelling of bone; Section 15 concerns the experimental validation of bone fracture simulations; Section 16 concerns bone fracture risk assessment. Lastly, glossaries for symbols, acronyms, and physico-mathematical terms are provided.

Extrafibrillar matrix yield stress and failure envelopes for mineralised collagen fibril arrays

Bone metabolic diseases such as osteoporosis constitute a major socio-economic challenge. A detailed understanding of the structure-property relationships of bone's underlying hierarchical levels has the potential to improve diagnosis and the ability to treat those diseases, especially with regards to the onset of failure. Therefore, elastic and yield properties of mineralised turkey leg tendon (MTLT), a mineralised tissue that is similar to bone but has a simpler multiscale structure, were investigated. Elastic properties were identified using a multiscale micromechanical model. The input parameters include constituent mechanical properties, volume fractions and inclusion aspect ratios and these were obtained from a wide variety of literature sources. The determined elastic properties were used to formulate micromechanically informed yield surfaces and to identify yield properties of MTLT at the nanometre length scale where failure is first reported to occur. This was done in conjunction with experimental results from the compression of micropillars extracted from individual mineralised collagen fibres. This data was then used to identify micromechanically informed failure envelopes. The shear yield stress of the extrafibrillar matrix, associated with interfibrillar sliding, was identified as 137.65 MPa. The ratio between tensile and compressive yield stress in the Drucker-Prager yield criterion was 0.65. For both criteria apparent yield stress of the mineralised collagen fibril decreased to 25.3-31.4% when varying fibril orientation from 0° to 90°. This study identified yield properties of extrafibrillar matrix using an aligned mineralised tissue. The ability to obtain yield stress data and unloading stiffness from micropillar compression tests of MTLT at the level of the mineralised collagen fibril array and downscaling these into the EM mitigates against possible errors associated with macroscopic stiffness predictions and proved to be an invaluable advantage compared to similar modelling approaches. Results may help to improve computational models that may then be used in pre-clinical testing or development of personalised treatment strategies.

Cellular and extracellular matrix of bone, with principles of synthesis and dependency of mineral deposition on cell membrane transport

Bone differs from other connective tissues; it is isolated by a layer of osteoblasts that are connected by tight and gap junctions. This allows bone to create dense lamellar type I collagen, control pH, mineral deposition, and regulate water content forming a compact and strong structure. New woven bone formed after degradation of mineralized cartilage is rapidly degraded and resynthesized to impart structural order for local bone strength. Ossification is regulated by thickness of bone units and by patterning via bone morphogenetic receptors including activin, other bone morphogenetic protein receptors, transforming growth factor-β receptors, all part of a receptor superfamily. This superfamily interacts with receptors for additional signals in bone differentiation. Important features of the osteoblast environment were established using recent tools including osteoblast differentiation in vitro. Osteoblasts deposit matrix protein, over 90% type I collagen, in lamellae with orientation alternating parallel or orthogonal to the main stress axis of the bone. Into this organic matrix, mineral is deposited as hydroxyapatite. Mineral matrix matures from amorphous to crystalline hydroxyapatite. This process includes at least two-phase changes of the calcium-phosphate mineral as well as intermediates involving tropocollagen fibrils to form the bone composite. Beginning with initiation of mineral deposition, there is uncertainty regarding cardinal processes, but the driving force is not merely exceeding the calcium-phosphate solubility product. It occurs behind a epithelial-like layer of osteoblasts, which generate phosphate and remove protons liberated during calcium-phosphate salt deposition. The forming bone matrix is discontinuous from the general extracellular fluid. Required adjustment of ionic concentrations and water removal from bone matrix are important details remaining to be addressed.

Insights into the kinetics of thermally induced crystallization of amorphous calcium phosphate

Transformations between amorphous and crystalline apatite mechanistically govern some of the most essential processes in bone metabolism, including biomineralization and bone remodeling. Fundamental understanding of this phase transition can help us gain control over the formation and dissolution of boney tissues in vivo and utilize that knowledge for various therapeutic ends. Crystallization of hydroxyapatite (HAp) and two tricalcium phosphate (TCP) polymorphs from the metastable precursor, amorphous calcium phosphate (ACP) was here studied kinetically and mechanistically using thermal analyses, X-ray diffraction and Fourier-transform infrared spectroscopy. Crystallization was detected in the differential thermal analysis as the exothermic peak at 639.5 °C at the slowest heating regimen of 5 °C min-1, while a combination of different kinetics models, including Augis-Bennett, Borchardt-Daniels, Johnson-Mehl-Avrami, Kissinger, Ozawa and Piloyan, yielded activation energies in the 435-450 kJ mol-1 range. Dehydrated ACP required a significant energy input to transform to HAp, thus indirectly proving the key role that structural water plays in this process in a biological setting. The phase transformation at high temperatures involved preformed nuclei and was solely due to their 3D growth, contrasting the edge-controlled nucleation derived earlier as the mechanism of growth in the solution. Crystallization was in both cases accompanied by the formation of needle-shape crystals of HAp through aggregation of ultrafine spherical units of ACP. Relationship between crystallinity and the heating rate was detected only for the initially amorphous structure, indicating a more intense and coherent lattice ordering process in annealed ACP than in HAp. Despite that, crystallization disobeyed the rule of inverse proportionality between the thermal energy required for the relaxation of defects and the level of strain, as the recovery rate of the initially poorly crystalline HAp was higher than that of ACP.

A homogenization approach for the effective drained viscoelastic properties of 2D porous media and an application for cortical bone

Closed-form solutions for the effective rheological properties of a 2D viscoelastic drained porous medium made of a Generalized Maxwell viscoelastic matrix and pore inclusions are developed and applied for cortical bone. The in-plane (transverse) effective viscoelastic bulk and shear moduli of the Generalized Maxwell rheology of the homogenized medium are expressed as functions of the porosity and the viscoelastic properties of the solid phase. When deriving these functions, the classical inverse Laplace-Carson transformation technique is avoided, due to its complexity, by considering the short and long term approximations. The approximated results are validated against exact solutions obtained from the inverse Laplace-Carson transform for a simple configuration when the later is available. An application for cortical bone with assumption of circular pore in the transverse plane shows that the proposed approximation fit very well with experimental data.

Multiscale modeling methods in biomechanics

More and more frequently, computational biomechanics deals with problems where the portion of physical reality to be modeled spans over such a large range of spatial and temporal dimensions, that it is impossible to represent it as a single space-time continuum. We are forced to consider multiple space-time continua, each representing the phenomenon of interest at a characteristic space-time scale. Multiscale models describe a complex process across multiple scales, and account for how quantities transform as we move from one scale to another. This review offers a set of definitions for this emerging field, and provides a brief summary of the most recent developments on multiscale modeling in biomechanics. Of all possible perspectives, we chose that of the modeling intent, which vastly affect the nature and the structure of each research activity. To the purpose we organized all papers reviewed in three categories: 'causal confirmation,' where multiscale models are used as materializations of the causation theories; 'predictive accuracy,' where multiscale modeling is aimed to improve the predictive accuracy; and 'determination of effect,' where multiscale modeling is used to model how a change at one scale manifests in an effect at another radically different space-time scale. Consistent with how the volume of computational biomechanics research is distributed across application targets, we extensively reviewed papers targeting the musculoskeletal and the cardiovascular systems, and covered only a few exemplary papers targeting other organ systems. The review shows a research subdomain still in its infancy, where causal confirmation papers remain the most common. WIREs Syst Biol Med 2017, 9:e1375. doi: 10.1002/wsbm.1375 For further resources related to this article, please visit the WIREs website.

Microscopic assessment of bone toughness using scratch tests

Bone is a composite material with five distinct structural levels: collagen molecules, mineralized collagen fibrils, lamellae, osteon and whole bone. However, most fracture testing methods have been limited to the macroscopic scale and there is a need for advanced characterization methods to assess toughness at the osteon level and below. The goal of this investigation is to present a novel framework to measure the fracture properties of bone at the microscopic scale using scratch testing. A rigorous experimental protocol is articulated and applied to examine cortical bone specimens from porcine femurs. The observed fracture behavior is very complex: we observe a strong anisotropy of the response with toughening mechanisms and a competition between plastic flow and brittle fracture. The challenge consists then in applying nonlinear fracture mechanics methods such as the J-integral or the energetic Size Effect Law to quantify the fracture toughness in a rigorous fashion. Our result suggests that mixed-mode fracture is instrumental in determining the fracture resistance. There is also a pronounced coupling between fracture and elasticity. Our methodology opens the door to fracture assessment at multiple structural levels, microscopic and potentially nanometer length scale, due to the scalability of scratch tests.

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We measure bone toughness at the osteon level using microscopic scratch tests.

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Scratch tests induce a ductile-to-brittle transition in cortical bone.

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Toughening mechanisms are observed at the microscopic scale.

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The J-integral and the size effect law are employed to predict the fracture energy.

Influence of interfacial interactions on deformation mechanism and interface viscosity in α-chitin-calcite interfaces

The interfaces between organic and inorganic phases in natural materials have a significant effect on their mechanical properties. This work presents a quantification of the interface stress as a function of interface chemical changes (water, organic molecules) in chitin-calcite (CHI-CAL) interfaces using classical non-equilibrium molecular dynamics (NEMD) simulations and steered molecular dynamics (SMD) simulations. NEMD is used to investigate interface stress as a function of applied strain based on the virial stress formulation. SMD is used to understand interface separation mechanism and to calculate interfacial shear stress based on a viscoplastic interfacial sliding model. Analyses indicate that interfacial shear stress combined with shear viscosity can result in variations to the mechanical properties of the examined interfacial material systems. It is further verified with Kelvin-Voigt and Maxwell viscoelastic analytical models representing viscous interfaces and outer matrix. Further analyses show that overall mechanical deformation depends on maximization of interface shear strength in such materials. This work establishes lower and upper bounds of interface strength in the interfaces examined.

The Role of Water Compartments in the Material Properties of Cortical Bone

Comprising ~20% of the volume, water is a key determinant of the mechanical behavior of cortical bone. It essentially exists in two general compartments: within pores and bound to the matrix. The amount of pore water-residing in the vascular-lacunar-canalicular space-primarily reflects intracortical porosity (i.e., open spaces within the matrix largely due to Haversian canals and resorption sites) and as such is inversely proportional to most mechanical properties of bone. Movement of water according to pressure gradients generated during dynamic loading likely confers hydraulic stiffening to the bone as well. Nonetheless, bound water is a primary contributor to the mechanical behavior of bone in that it is responsible for giving collagen the ability to confer ductility or plasticity to bone (i.e., allows deformation to continue once permanent damage begins to form in the matrix) and decreases with age along with fracture resistance. Thus, dehydration by air-drying or by solvents with less hydrogen bonding capacity causes bone to become brittle, but interestingly, it also increases stiffness and strength across the hierarchical levels of organization. Despite the importance of matrix hydration to fracture resistance, little is known about why bound water decreases with age in hydrated human bone. Using (1)H nuclear magnetic resonance (NMR), both bound and pore water concentrations in bone can be measured ex vivo because the proton relaxation times differ between the two water compartments, giving rise to two distinct signals. There are also emerging techniques to measure bound and pore water in vivo with magnetic resonance imaging (MRI). The NMR/MRI-derived bound water concentration is positively correlated with both the strength and toughness of hydrated bone and may become a useful clinical marker of fracture risk.

The Role of Hydroxyl Channel in Defining Selected Physicochemical Peculiarities Exhibited by Hydroxyapatite

Mysteries surrounding the most important mineral for the vertebrate biology, hydroxyapatite, are many. Perhaps the Greek root of its name, απαταo, meaning 'to deceive' and given to its mineral form by the early gem collectors who confused it with more precious stones, is still applicable today, though in a different connotation, descriptive of a number of physicochemical peculiarities exhibited by it. Comparable to water as the epitome of peculiarities in the realm of liquids, hydroxyapatite can serve as a paradigm for peculiarities in the world of solids. Ten of the peculiar properties of hydroxyapatite are sketched in this review piece, ranging from (i) the crystal lattice flexibility to (ii) notorious surface layer instability to (iii) finite piezoelectricity, pyroelectricity and conductivity to protons to (iv) accelerated growth and improved osteoconductivity in the electromagnetic fields to (v) high nucleation rate at low supersaturations and low crystal growth rate at high supersaturations to (vi) higher bioactivity and resorbability of biological apatite compared to the synthetic ones, and beyond. An attempt has been made to explain this array of curious characteristics by referring to a particular element of the crystal structure of hydroxyapatite: the hydroxyl ion channel extending in the direction of the c-axis, through a crystallographic column created by the overlapping calcium ion triangles.

Poromicromechanics reveals that physiological bone strains induce osteocyte-stimulating lacunar pressure

Mechanical loads which are macroscopically acting onto bony organs, are known to influence the activities of biological cells located in the pore spaces of bone, in particular so the signaling and production processes mediated by osteocytes. The exact mechanisms by which osteocytes are actually able to “feel” the mechanical loading and changes thereof, has been the subject of numerous studies, and, while several hypotheses have been brought forth over time, this topic has remained a matter of debate. Relaxation times reported in a recent experimental study of Gardinier et al. (Bone 46(4):1075–1081, 2010) strongly suggest that the lacunar pores are likely to experience, during typical physiological load cycles, not only fluid transport, but also undrained conditions. The latter entail the buildup of lacunar pore pressures, which we here quantify by means of a thorough multiscale modeling approach. In particular, the proposed model is based on classical poroelasticity theory, and able to account for multiple pore spaces. First, the model reveals distinct nonlinear dependencies of the resulting lacunar (and vascular) pore pressures on the underlying bone composition, highlighting the importance of a rigorous multiscale approach for appropriate computation of the aforementioned pore pressures. Then, the derived equations are evaluated for macroscopic (uniaxial as well as hydrostatic) mechanical loading of physiological magnitude. The resulting model-predicted pore pressures agree very well with the pressures that have been revealed, by means of in vitro studies, to be of adequate magnitude for modulating the responses of biological cells, including osteocytes. This underlines that osteocytes may respond to many types of loading stimuli at the same time, in particular so to fluid flow and hydrostatic pressure.

The nano-epsilon dot method for strain rate viscoelastic characterisation of soft biomaterials by spherical nano-indentation

Nano-indentation is widely used for probing the micromechanical properties of materials. Based on the indentation of surfaces using probes with a well-defined geometry, the elastic and viscoelastic constants of materials can be determined by relating indenter geometry and measured load and displacement to parameters which represent stress and deformation. Here we describe a method to derive the viscoelastic properties of soft hydrated materials at the micro-scale using constant strain rates and stress-free initial conditions. Using a new self-consistent definition of indentation stress and strain and corresponding unique depth-independent expression for indentation strain rate, the epsilon dot method, which is suitable for bulk compression testing, is transformed to nano-indentation. We demonstrate how two materials can be tested with a displacement controlled commercial nano-indentor using the nano-espilon dot method (nano-ε̇M) to give values of instantaneous and equilibrium elastic moduli and time constants with high precision. As samples are tested in stress-free initial conditions, the nano-ε̇M could be useful for characterising the micro-mechanical behaviour of soft materials such as hydrogels and biological tissues at cell length scales.

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.

A multiscale poromicromechanical approach to wave propagation and attenuation in bone

Ultrasonics is an important diagnostic tool for bone diseases, as it allows for non-invasive assessment of bone tissue quality through mass density-elasticity relationships. The latter are, however, quite complex for fluid-filled porous media, which motivates us to develop a rigorous multiscale poromicrodynamics approach valid across the great variety of different bone tissues. Multiscale momentum and mass balance, as well as kinematics of a hierarchical double porous medium, together with Darcy's law for fluid flow and micro-poro-elasticity for the solid phase of bone, give access to the so-called dispersion relation, linking the complex wave numbers to corresponding wave frequencies. Experimentally validated results show that 2.25 MHz acoustical signals transmit healthy cortical bone (exhibiting a low vascular porosity) only in the form of fast waves, agreeing very well with experimental data, while both fast and slow waves transmit highly osteoporotic as well as trabecular bone (exhibiting a large vascular porosity). While velocities and wavelengths of both fast and slow waves, as well as attenuation lengths of slow waves, are always monotonously increasing with the permeability of the bone sample, the attenuation length of fast waves shows a minimum when considered as function of the permeability.

Fibrillar structure and elasticity of hydrating collagen: a quantitative multiscale approach

It is well known that hydration of collagenous tissues leads to their swelling, as well as to softening of their elastic behavior. However, it is much less clear which microstructural and micromechanical "rules" are involved in this process. Here, we develop a theoretical approach cast in analytical mathematical formulations, which is experimentally validated by a wealth of independent tests on collagenous tissues, such as X-ray diffraction, vacuum drying, mass measurements, and Brillouin light scattering. The overall emerging picture is the following: air-drying leaves water only in the gap zones between the triple-helical collagen molecules; upon re-hydration, the extrafibrillar space is established at volumes directly proportional to the hydration-induced swelling of the (micro) fibrils, until the maximum equatorial distance between the long collagen molecules is reached. Thereafter, the volume of the fibrils stays constant, and only the extrafibrillar volume continues to grow. At all these hydration stages, the elastic behavior is governed by the same, hydration-invariant mechanical interaction pattern of only two, interpenetrating mechanical phases: transversely isotropic molecular collagen and isotropic water (or empty pores in the vacuum-dried case).