Hot off the presses! Jan 11 J Biomech

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Latest Articles Include:

• Editorial Board and Publication Information
  - J Biomech 44(2):IFC (2011)
  
• Bone tissue: Hierarchical simulations for clinical applications
  - J Biomech 44(2):211-212 (2011)
  
• Challenges to bone formation in spinal fusion
  - J Biomech 44(2):213-220 (2011)
  Spinal arthrodesis continues to expand in clinical indications and surgical practice. Despite a century of study, failure of bone formation or pseudarthrosis can occur in individual patients with debilitating clinical symptoms. Here we review biological and technical aspects of spinal fusion under active investigation, describe relevant biomechanics in health and disease, and identify the possibilities and limitations of translational animal models. The purpose of this article is to foster collaborative efforts with researchers who model bone hierarchy. The induction of heterotopic osteosynthesis requires a complex balance of biologic factors and operative technique to achieve successful fusion. Anatomical considerations of each spinal region including blood supply, osteology, and biomechanics predispose a fusion site to robust or insufficient bone formation. Careful preparation of the fusion site and appropriate selection of graft materials remains critical but is som! etimes guided by conflicting evidence from the long-bone literature. Modern techniques of graft site preparation and instrumentation have evolved for every segment of the vertebral column. Despite validated biomechanical studies of modern instrumentation, a correlation with superior clinical outcomes is difficult to demonstrate. In many cases, adjuvant biologic therapies with allograft and synthetic cages have been used successfully to reproduce the enhancement of fusion rates observed with cancellous and tricortical autograft. Current areas of investigation comprise materials science, stem cell therapies, recombinant growth factors, scaffolds and biologic delivery systems, and minimally invasive surgical techniques to optimize the biologic response to intervention. Diverse animal models are required to approach the breadth of spinal pathology and novel therapeutics.
• Quality of intertrochanteric cancellous bone as predictor of femoral stem RSA migration in cementless total hip arthroplasty
  - J Biomech 44(2):221-227 (2011)
  In cementless total hip arthroplasty, osteoporosis may jeopardize the achievement of immediate stability and lead to migration of anatomically shaped femoral stems. Poor quality of proximal cancellous bone per se may also affect the rate of osseointegration. In a selected group of female total hip arthroplasty patients (mean age 64 years) with unremarkable medical history, intertrochanteric cancellous bone biopsy was taken from the site of stem implantation. Local bone quality, determined by structural μCT imaging and destructive compression testing of the biopsy tissue, was used as the predictor of three-dimensional stem migration determined by radiostereometric analysis (RSA) up to 24 months. The patients exhibited major differences in mechanical properties of the intertrochanteric cancellous bone, which were closely related to the structural parameters calculated from μCT data. Unexpectedly, the major differences observed in the quality of trochanteric cancellous ! bone had only minor reflections in the RSA migration of the femoral stems. In statistical analysis, the μCT-based bone mineral density quartile (low, middle, high) was the only significant predictor for stem translation at 24 months (p=0.022) but only a small portion (R2=0.16) of the difference in translation could be explained by changes in bone mineral density quartile. None of the other parameters investigated predicted stem migration in translation or rotation. In conclusion, poor quality of intertrochanteric cancellous bone seems to contribute to the risk of implant migration less than expected. Probably also the importance of surgical preservation of intertrochanteric cancellous bone has been over-emphasized for osseointegration of cementless stem.
• The behavior of the micro-mechanical cement–bone interface affects the cement failure in total hip replacement
  - J Biomech 44(2):228-234 (2011)
  In the current study, the effects of different ways to implement the complex micro-mechanical behavior of the cement–bone interface on the fatigue failure of the cement mantle were investigated. In an FEA-model of a cemented hip reconstruction the cement–bone interface was modeled and numerically implemented in four different ways: (I) as infinitely stiff, (II) as infinitely strong with a constant stiffness, (III) a mixed-mode failure response with failure in tension and shear, and (IV) realistic mixed mode behavior obtained from micro-FEA models. Case II, III, and IV were analyzed using data from a stiff and a compliant micro-FEA model and their effects on cement failure were analyzed. The data used for Case IV was derived from experimental specimens that were tested previously. Although the total number of cement cracks was low for all cases, the compliant Case II resulted in twice as many cracks as Case I. All cases caused similar stress distributions at the int! erface. In all cases, the interface did not display interfacial softening; all stayed the elastic zone. Fatigue failure of the cement mantle resulted in a more favorable stress distribution at the cement–bone interface in terms of less tension and lower shear tractions. We conclude that immediate cement–bone interface failure is not likely to occur, but its local compliancy does affect the formation of cement cracks. This means that at a macro-level the cement–bone interface should be modeled as a compliant layer. However, implementation of interfacial post-yield softening does seems to be necessary.
• Challenges in relating experimental hip implant fixation predictions to clinical observations
  - J Biomech 44(2):235-243 (2011)
  Long-term clinical follow-up studies have shown that radiolucent lines at the cement interfaces of total hip replacement femoral components develop gradually, ultimately leading to loosening. In this experimental study, 32 synthetic femurs implanted with cemented femoral components were cyclically loaded with a dynamic joint reaction force, torque, and muscle force, to assess the relative effects of surface finish and collars on interface fixation. Four each of four otherwise identical straight femoral stems, varying only in surface finish and presence or lack of collars were used. Specimens were tested under two conditions: (1) with intact interfaces simulating immediate post-operative conditions and (2) with a thin-film at the stem–cement interface, simulating conditions several weeks to months post-operative when fibrous tissue has formed with the implant still stable. Micromotion was measured at both interfaces in three directions. Surface finish had a larger rel! ative effect than collars, regardless of whether or not a thin-film was present. For example, a proximal grit-blasted finish enhanced fixation at the stem–cement interface by 7–12 μm per-cycle (p<0.05) and decreased early cement mantle loosening by 7–13 μm. For straight stems, rougher surfaces provided greater stability than polished, even with a thin film at the stem–cement interfaces, contradicting the theory that once debonded, rough stems are less stable than polished at the stem–cement interface. The findings of this experimental study exemplify the need to take advantage of all available tools for the preclinical evaluation of orthopaedic implants, including long-term clinical observations of related devices, analytical and numeric models, and experimental bench-top simulations.
• Metabolic bone disease: Atypical femoral fractures
  - J Biomech 44(2):244-247 (2011)
  Since 2005 reports have been published describing unusual femoral shaft fractures primarily in postmenopausal women treated for prolonged periods with a bisphosphonate drug for osteoporosis. In some patients pain develops in the femur prior to a completed fracture. Bilateral fractures have occurred in some patients. It is unclear whether oversuppression of bone cell activity is a major factor in the pathogenesis of the fractures, or whether these are a rare manifestation of the underlying bone disease. Such fractures do occur in other metabolic bone disorders in which there are marked abnormalities of bone structure.
• Variation of trabecular architecture in proximal femur of postmenopausal women
  - J Biomech 44(2):248-256 (2011)
  This investigation of microstructure in the human proximal femur probes the relationship between the parameters of the FRAX index of fracture risk and the parameters of bone microstructure. The specificity of fracture sites at the proximal femur raises the question of whether trabecular parameters are site-specific during post-menopause, before occurrence of fragility fracture. The donated proximal femurs of sixteen post-menopausal women in the sixth and seventh decades of life, free of metabolic pathologies and therapeutic interventions that could have altered the bone tissue, constituted the material of the study. We assessed bone mineral density of the proximal femurs by dual energy X-ray absorptiometry and then sectioned the femurs through the center of the femoral head and along the femoral neck axis. For each proximal femur, morphometry of trabeculae was conducted on the plane of the section divided into conventional regions and sub-regions consistent with the pr! eviously identified trabecular families that provide regions of relatively homogeneous microstructure. Mean trabecular width and percent bone area were calculated at such sites. Our findings indicate that each of mean trabecular width and percent bone area vary within each proximal femur independently from each other, with dependence on site. Both trabecular parameters show significant differences between pairs of sites. We speculate that a high FRAX index at the hip corresponds to a reduced percent bone area among sites that gives a more homogeneous and less site-specific quality to the proximal femur. This phenomenon may render the local tissue less able to carry out the expected mechanical function.
• Variations in morphological and biomechanical indices at the distal radius in subjects with identical BMD
  - J Biomech 44(2):257-266 (2011)
  Determination of osteoporotic status is based primarily on areal bone mineral density (aBMD) obtained through dual X-ray absorptiometry (DXA). However, many fractures occur in patients with T-scores above the WHO threshold of osteoporosis, in part because DXA measures are insensitive to biomechanically important alterations in bone quality. The goal of this study was to determine – within groups of subjects with identical radius aBMD values – the extant variation in densitometric, geometric, microstructural, and biomechanical parameters. High resolution peripheral quantitative computed tomography (HR-pQCT) and DXA radius data from males and females spanning large ranges in age, osteoporotic status, and anthropometrics were compiled. 262 distal radius datasets were processed for this study. HR-pQCT scans were analyzed according to the manufacturer's standard clinical protocol to quantify densitometric, geometric, and microstructural indices. Micro-finite element a! nalysis was performed to calculate biomechanical indices. Factor of risk of wrist fracture was calculated. Simulated aBMD calculated from HR-pQCT data was used to group scans for evaluation of variation in quantified indices. Indices reflecting the greatest variation within aBMD level were BMD in the central portion of the trabecular compartment (max CV 142), trabecular heterogeneity (max CV 90), and intra-cortical porosity (max CV 151). Of the biomechanical indices, cortical load fraction had the greatest variation (max CV 38). Substantial variations in indices reflecting density, structure, and biomechanical competence exist among subjects with identical aBMD levels. Overlap of these indices among osteoporotic status groups reflects the reported incidence of osteoporotic fracture in subjects classified as osteopenic or normal.
• Extent and location of bone loss at dental implants in patients with peri-implantitis
  - J Biomech 44(2):267-271 (2011)
  Peri-implantitis is an infectious disease, which leads to loss of supporting bone around dental implants. To evaluate the extent and location of bone loss, 43 patients with peri-implantitis were examined. The bone loss was clinically measured at the time of dental surgery. Data revealed that 25% of subjects had bone loss associated with all their implants although the majority of the subjects had fewer than 50% of their implants affected by bone loss. A total number of 264 implants were examined and 131 of those had peri-implantitis associated bone loss. The pattern of bone loss at implants varied between and within subjects and location in the jaws. The highest proportion of implants with peri-implantitis was found in the upper jaw and within this group, at implants located in the incisor area of the upper jaw; the lowest was the canine area of the lower jaw. The highest proportion of implants that lost ≥2/3 of their bone support was found in the incisor area of the! maxilla. We concluded that in the presence of peri-implant inflammation, bone quantity and characteristics may influence the progression of peri-implantitis bone loss at dental implants. We hypothesize that the ability of the bone to withstand occlusal forces will be altered as consequence of the loss of bone at the neck of the implants. To achieve an understanding of the local degradation of bone due to peri-implantitis, we need to analyze the microstructure of the bone as well the cellular biology of the peri-implant inflammation.
• Determination of dynamically adapting anisotropic material properties of bone under cyclic loading
  - J Biomech 44(2):272-276 (2011)
  Because bone tissue adapts to loading conditions, finite element simulations of remodelling bone require a precise prediction of dynamically changing anisotropic elastic parameters. We present a phenomenological theory that refers to the tissue in terms of the tendency of the structure to align with principal stress directions. We describe the material parameters of remodelling bone. This work follows findings by the same research group and independently by Danilov (1971) in the field of plasticity, where the dependencies of the components of the stiffness tensor in terms of time are based on Hill's anisotropy. We modify such an approach in this novel theory that addresses bone tissue that can regenerate. The computational assumption of the theory is that bone trabeculae have the tendency to orient along one of the principal stress directions but during remodelling the principal stresses change continuously and the resulting orientation of the trabeculae can differ f! rom the principal stress direction at any given time. The novelty of this work consists in the limited number of parameters needed to compute the twenty-one anisotropic material parameters at any given location in the bone tissue. In addition to the theory, we present here two cases of simplified geometry, loading and boundary conditions to show the effect of (1) time on the material properties; and (2) change of loading conditions on the anisotropic parameters. The long term goal is to experimentally verify that the predictions generated by theory provide a reliable simulation of cancellous bone properties.
• Microstructure and nanomechanical properties in osteons relate to tissue and animal age
  - J Biomech 44(2):277-284 (2011)
  Material property changes in bone tissue with ageing are a crucial missing component in our ability to understand and predict age-related fracture. Cortical bone osteons contain a natural gradient in tissue age, providing an ideal location to examine these effects. This study utilized osteons from baboons aged 0–32 years (n=12 females), representing the baboon lifespan, to examine effects of tissue and animal age on mechanical properties and composition of the material. Tissue mechanical properties (indentation modulus and hardness), composition (mineral-to-matrix ratio, carbonate substitution, and crystallinity), and aligned collagen content (aligned collagen peak height ratio) were sampled along three radial lines in three osteons per sample by nanoindentation, Raman spectroscopy, and second harmonic generation microscopy, respectively. Indentation modulus, hardness, mineral-to-matrix ratio, carbonate substitution, and aligned collagen peak height ratio followed bi! phasic relationships with animal age, increasing sharply during rapid growth before leveling off at sexual maturity. Mineral-to-matrix ratio and carbonate substitution increased 12% and 6.7%, respectively, per year across young animals during growth, corresponding with a nearly 7% increase in stiffness and hardness. Carbonate substitution and aligned collagen peak height ratio both increased with tissue age, increasing 6–12% across the osteon radii. Indentation modulus most strongly correlated with mineral-to-matrix ratio, which explained 78% of the variation in indentation modulus. Overall, the measured compositional and mechanical parameters were the lowest in tissue of the youngest animals. These results demonstrate that composition and mechanical function are closely related and influenced by tissue and animal age.
• Influences of spherical tip radius, contact depth, and contact area on nanoindentation properties of bone
  - J Biomech 44(2):285-290 (2011)
  Nanoindentation has been widely used as a means to measure the micro-mechanical properties of bone and to predict the macroscopic properties. The role of indent depth and indenter tip geometry in measuring the hierarchical properties of bone tissue was explored experimentally using a range of spherical indenter tips of R=5, 25, 65, and 200 μm. Nanoindentation arrays, not targeted to fall on specific structures or locations, enabled statistical sampling of osteons within PMMA-embedded, bovine, cortical bone on a single sample to a range of maximum displacements (minimum of 100 nm and maximum of 2000 nm). Elastic finite element models were then utilized to isolate the contributions of indenter tip radius, contact area, and position within the lamellar structure in comparison to the experimental results. For a small, R=5 μm indenter tip, indentation modulus consistently increased with contact depth and increased plastic deformation, resulting in an artificial increase i! n elastic properties. While larger radius tips (R=25, 65, and 200 μm) did not enable evaluation of a high spatial resolution on the surface, they produced data that was representative of the lower load and contact depth measurements with the smaller tip. However the sensitivity to mechanical property variations across the 2-D surface of the material was lost with increase in indenter tip size. Correspondingly, measurement variance was also decreased as the volume contributing to the indent response represented an average of surface roughness, varying mineral content, defects, and underlying tissue type and structure.
• Internal strain gradients quantified in bone under load using high-energy X-ray scattering
  - J Biomech 44(2):291-296 (2011)
  High-energy synchrotron X-ray scattering (>60 keV) allows noninvasive quantification of internal strains within bone. In this proof-of-principle study, wide angle X-ray scattering maps internal strain vs position in cortical bone (murine tibia, bovine femur) under compression, specifically using the response of the mineral phase of carbonated hydroxyapatite. The technique relies on the response of the carbonated hydroxyapatite unit cells and their Debye cones (from nanocrystals correctly oriented for diffraction) to applied stress. Unstressed, the Debye cones produce circular rings on the two-dimensional X-ray detector while applied stress deforms the rings to ellipses centered on the transmitted beam. Ring ellipticity is then converted to strain via standard methods. Strain is measured repeatedly, at each specimen location for each applied stress. Experimental strains from wide angle X-ray scattering and an attached strain gage show bending of the rat tibia and agree ! qualitatively with results of a simplified finite element model. At their greatest, the apatite-derived strains approach 2500 με on one side of the tibia and are near zero on the other. Strains maps around a hole in the femoral bone block demonstrate the effect of the stress concentrator as loading increased and agree qualitatively with the finite element model. Experimentally, residual strains of approximately 2000 με are present initially, and strain rises to approximately 4500 με at 95 MPa applied stress (about 1000 με above the strain in the surrounding material). The experimental data suggest uneven loading which is reproduced qualitatively with finite element modeling.
• Raman and mechanical properties correlate at whole bone- and tissue-levels in a genetic mouse model
  - J Biomech 44(2):297-303 (2011)
  The fracture resistance of bone arises from the composition, orientation, and distribution of the primary constituents at each hierarchical level of organization. Therefore, to establish the relevance of Raman spectroscopy (RS) in identifying differences between strong or tough bone and weak or brittle bone, we investigated whether Raman-derived properties could explain the variance in biomechanical properties at both the whole bone and the tissue-level, and do so independently of traditional measurements of mineralization. We harvested femurs from wild-type mice and mice lacking matrix metalloproteinase 2 because the mutant mice have a known reduction in mineralization. Next, RS quantified compositional properties directly from the intact diaphysis followed by micro-computed tomography to quantify mineralization density (Ct.TMD). Correlations were then tested for significance between these properties and the biomechanical properties as determined by the three-point be! nding test on the same femurs. Harvested tibia were embedded in plastic, sectioned transversely, and polished in order to acquire average Raman properties per specimen that were then correlated with average nanoindentation properties per specimen. Dividing the ν1 phosphate by the proline peak intensity provided the strongest correlation between the mineral-to-collagen ratio and the biomechanical properties (whole bone modulus, strength, and post-yield deflection plus nanoindentation modulus). Moreover, the linear combination of ν1 phosphate/proline and Ct.TMD provided the best explanation of the variance in strength between the genotypes, and it alone was the best explanatory variable for brittleness. Causal relationships between Raman and fracture resistance need to be investigated, but Raman has the potential to assess fracture risk.
• Top down and bottom up engineering of bone
  - J Biomech 44(2):304-312 (2011)
  The goal of this retrospective article is to place the body of my lab′s multiscale mechanobiology work in context of top-down and bottom-up engineering of bone. We have used biosystems engineering, computational modeling and novel experimental approaches to understand bone physiology, in health and disease, and across time (in utero, postnatal growth, maturity, aging and death, as well as evolution) and length scales (a single bone like a femur, m; a sample of bone tissue, mm–cm; a cell and its local environment, μm; down to the length scale of the cell's own skeleton, the cytoskeleton, nm). First we introduce the concept of flow in bone and the three calibers of porosity through which fluid flows. Then we describe, in the context of organ–tissue, tissue–cell and cell–molecule length scales, both multiscale computational models and experimental methods to predict flow in bone and to understand the flow of fluid as a means to deliver chemical and mechanical c! ues in bone. Addressing a number of studies in the context of multiple length and time scales, the importance of appropriate boundary conditions, site specific material parameters, permeability measures and even micro-nanoanatomically correct geometries are discussed in context of model predictions and their value for understanding multiscale mechanobiology of bone. Insights from these multiscale computational modeling and experimental methods are providing us with a means to predict, engineer and manufacture bone tissue in the laboratory and in the human body.
• Multi-scale characterization of swine femoral cortical bone
  - J Biomech 44(2):313-320 (2011)
  Multi-scale experimental work was carried out to characterize cortical bone as a heterogeneous material with hierarchical structure, which spans from nanoscale (mineralized collagen fibril), sub-microscale (single lamella), microscale (lamellar structures), to mesoscale (cortical bone) levels. Sections from femoral cortical bone from 6, 12, and 42 months old swines were studied to quantify the age-related changes in bone structure, chemical composition, and mechanical properties. The structural changes with age from sub-microscale to mesoscale levels were investigated with scanning electron microscopy and micro-computed tomography. The chemical compositions at mesoscale were studied by ash content method and dual energy X-ray absorptiometry, and at microscale by Fourier transform infrared microspectroscopy. The mechanical properties at mesoscale were measured by tensile testing, and elastic modulus and hardness at sub-microscale were obtained using nanoindentation. The! experimental results showed age-related changes in the structure and chemical composition of cortical bone. Lamellar bone was a prevalent structure in 6 months and 12 months old animals, resorption sites were most pronounced in 6 months old animals, while secondary osteons were the dominant features in 42 months old animals. Mineral content and mineral-to-organic ratio increased with age. The structural and chemical changes with age corresponded to an increase in local elastic modulus, and overall elastic modulus and ultimate tensile strength as bone matured.
• Multiscale modeling of bone tissue with surface and permeability control
  - J Biomech 44(2):321-329 (2011)
  Natural biological materials usually present a hierarchical arrangement with various structural levels. The biomechanical behavior of the complex hierarchical structure of bone is investigated with models that address the various levels corresponding to different scales. Models that simulate the bone remodeling process concurrently at different scales are in development. We present a multiscale model for bone tissue adaptation that considers the two top levels, whole bone and trabecular architecture. The bone density distribution is calculated at the macroscale (whole bone) level, and the trabecular structure at the microscale level takes into account its mechanical properties as well as surface density and permeability. The bone remodeling process is thus formulated as a material distribution problem at both scales. At the local level, the biologically driven information of surface density and permeability characterizes the trabecular structure. The model is tested by! a three-dimensional simulation of bone tissue adaptation for the human femur. The density distribution of the model shows good agreement with the actual bone density distribution. Permeability at the microstructural level assures interconnectivity of pores, which mimics the interconnectivity of trabecular bone essential for vascularization and transport of nutrients. The importance of this multiscale model relays on the flexibility to control the morphometric parameters that characterize the trabecular structure. Therefore, the presented model can be a valuable tool to define bone quality, to assist with diagnosis of osteoporosis, and to support the development of bone substitutes.
• The relative contributions of non-enzymatic glycation and cortical porosity on the fracture toughness of aging bone
  - J Biomech 44(2):330-336 (2011)
  The risk of fracture increases with age due to the decline of bone mass and bone quality. One of the age-related changes in bone quality occurs through the formation and accumulation of advanced glycation end-products (AGEs) due to non-enzymatic glycation (NEG). However as a number of other changes including increased porosity occur with age and affect bone fragility, the relative contribution of AGEs on the fracture resistance of aging bone is unknown. Using a high-resolution nonlinear finite element model that incorporate cohesive elements and micro-computed tomography-based 3d meshes, we investigated the contribution of AGEs and cortical porosity on the fracture toughness of human bone. The results show that NEG caused a 52% reduction in propagation fracture toughness (R-curve slope). The combined effects of porosity and AGEs resulted in an 88% reduction in propagation toughness. These findings are consistent with previous experimental results. The model captured th! e age-related changes in the R-curve toughening by incorporating bone quantity and bone quality changes, and these simulations demonstrate the ability of the cohesive models to account for the irreversible dynamic crack growth processes affected by the changes in post-yield material behavior. By decoupling the matrix-level effects due to NEG and intracortical porosity, we are able to directly determine the effects of NEG on fracture toughness. The outcome of this study suggests that it may be important to include the age-related changes in the material level properties by using finite element analysis towards the prediction of fracture risk.
• Spectral analysis and connectivity of porous microstructures in bone
  - J Biomech 44(2):337-344 (2011)
  Cancellous bone is a porous composite of calcified tissue interspersed with soft marrow. Sea ice is also a porous composite, consisting of pure ice with brine, air, and salt inclusions. Interestingly, the microstructures of bone and sea ice exhibit notable similarities. In recent years, we have developed mathematical and experimental techniques for imaging and characterizing the brine microstructure of sea ice, such as its volume fraction and connectivity, as well as a range of theoretical approaches for studying fluid, thermal, and electromagnetic transport in sea ice. Here we explore the application of our sea ice techniques to investigate trabecular bone. For example, percolation theory that quantifies brine connectivity and its thermal evolution can also help assess the impact of osteoporosis on trabecular structure. Central to our approach is the spectral measure of a composite material, which contains detailed information about the mixture geometry, and can be us! ed in powerful integral representations to compute the effective properties. The spectral measure is obtained from the eigenvalues and eigenvectors of a self-adjoint operator determined exclusively by the composite microgeometry. Here we compute the spectral measures for discretizations of images of healthy and osteoporotic bone. The measures are used to compute the effective electromagnetic properties of the bone specimens. These data are then inverted to reconstruct the porosity of the original specimens, with excellent agreement.
• Characterization of structure and properties of bone by spectral measure method
  - J Biomech 44(2):345-351 (2011)
  Novel mathematical method called spectral measure method (SMM) is developed for characterization of bone structure and indirect estimation of bone properties. The spectral measure method is based on an inverse homogenization technique which allows to derive information about the structure of composite material from measured effective electric or viscoelastic properties. The mechanical properties and ability to withstand fracture depend on the structural organization of bone as a hierarchical composite. Information about the bone structural parameters is contained in the spectral measure in the Stieltjes integral representation of the effective properties. The method is based on constructing the spectral measure either by calculating it directly from micro-CT images or using measurements of electric or viscoelastic properties over a frequency range. In the present paper, we generalize the Stieltjes representation to the viscoelastic case and show how bone microstructure! , in particular, bone volume or porosity, can be characterized by the spectral function calculated using measurements of complex permittivity or viscoelastic modulus. For validation purposes, we numerically simulated measured data using micro-CT images of cancellous bone. Recovered values of bone porosity are in excellent agreement with true porosity estimated from the micro-CT images. We also discuss another application of this method, which allows to estimate properties difficult to measure directly. The spectral measure method based on the derived Stieltjes representation for viscoelastic composites, has a potential for non-invasive characterization of bone structure using electric or mechanical measurements. The method is applicable to sea ice, porous rock, and other composite materials.
• Wavelet decomposition of transmitted ultrasound wave through a 1-D muscle–bone system
  - J Biomech 44(2):352-358 (2011)
  In the attempt for using ultrasound as a diagnostic device for osteoporosis, several authors have described the result of the in vitro experiment in which ultrasound is passed through a cancellous bone specimen placed in a water tank. However, in the in vivo setting, a patient's cancellous bone is surrounded by cortical and muscle layers. This paper considers in the one-dimensional case (1) what effect the cortical bone segments surrounding the cancellous segment would have on the received signal and (2) what the received signal would be when a source and receiver are placed on opposite sides of a structure consisting of a cancellous segment surrounded by cortical and muscle layers. Mathematically this is accomplished by representing the received signal as a sum of wavelets which go through different reflection–transmission histories at the muscle–cortical bone and cortical-cancellous bone interfaces. The muscle and cortical bone are modeled as elastic materials an! d the cancellous bone as a poroelastic material described by the Biot–Johnson–Koplik–Dashen model. The approach presented here permits the assessment of which possible paths of transmission and reflection through the cortical-cancellous or muscle-cortical-cancellous complex will result in significant contributions to the received waveform. This piece of information can be useful for solving the inverse problem of non-destructive assessment of material properties of bone. Our methodology can be generalized to three-dimensional parallelly layered structure by first applying Fourier transform in the directions perpendicular to the transverse direction.
• Chemotaxis of mesenchymal stem cells within 3D biomimetic scaffolds—a modeling approach
  - J Biomech 44(2):359-364 (2011)
  Bone tissue engineering is a promising strategy to repair local defects by implanting biodegradable scaffolds which undergo remodeling and are replaced completely by autologous bone tissue. Here, we consider a Keller–Segel model to describe the chemotaxis of bone marrow-derived mesenchymal stem cells (BMSCs) into a mineralized collagen scaffold. Following recent experimental results in bone healing, demonstrating that a sub-population of BMSCs can be guided into 3D scaffolds by gradients of signaling molecules such as SDF-1α, we consider a population of BMSCs on the surface of the pore structure of the scaffold and the chemoattractant SDF-1α within the pores. The resulting model is a coupled bulk/surface model which we reformulate following a diffuse-interface approach in which the geometry is implicitly described using a phase-field function. We explain how to obtain such an implicit representation and present numerical results on μCT-data for real scaffolds, ass! uming a diffusion of SDF-1α being coupled to diffusion and chemotaxis of the cells towards SDF-1α. We observe a slowing-down of BMSC ingrowth after the scaffold becomes saturated with SDF-1α, suggesting that a slow release of SDF-1α avoiding an early saturation is required to enable a complete colonization of the scaffold. The validation of our results is possible via SDF-1α release from injectible carrier materials, and an adaption of our model to similar coupled bulk/surface problems such as remodeling processes seems attractive.