Cellular scale model of growth plate: An in silico model of chondrocyte hypertrophy.

The growth plate is the responsible for longitudinal bone growth. It is a cartilaginous structure formed by chondrocytes that are continuously undergoing a differentiation process that starts with a highly proliferative state, followed by cellular hypertrophy, and finally tissue ossification. Within the growth plate chondrocytes display a characteristic columnar organization that potentiates longitudinal growth. Both chondrocyte organization and hypertrophy are highly regulated processes influenced by biochemical and mechanical stimuli. These processes have been studied mainly using in vivo models, although there are few computational approaches focused on the rate of ossification rather than events at cellular level. Here, we developed a model of cellular behavior integrating biochemical and structural factors in a single column of cells in the growth plate. In our model proliferation and hypertrophy were controlled by biochemical regulatory loop formed between Ihh and PTHrP (modeled as a set of reaction-diffusion equations), while cell growth was controlled by mechanical loading. We also examined the effects of static loading. The model reproduced the proliferation and hypertrophy of chondrocytes in organized columns. This model constitutes a first step towards the development of mechanobiological models that can be used to study biochemical interactions during endochondral ossification.

Prolonging disuse in aged mice amplifies cortical but not trabecular bones' response to mechanical loading.

OBJECTIVE: Short-term neurectomy-induced disuse (SN) has been shown to restore load responses in aged mice. We examined whether this restoration was further enhanced in both cortical and trabecular bone by simply extending the SN. METHODS: Following load:strain calibration, tibiae in female C57BL/J6 mice at 8, 14 and 20 weeks and 18 months (n=8/group) were loaded and bone changes measured. Effects of long-term SN examined in twenty-six 18 months-old mice, neurectomised for 5 or 100 days with/without subsequent loading. Cortical and trabecular responses were measured histomorphometrically or by micro-computed tomography. RESULTS: Loading increased new cortical bone formation, elevating cross-sectional area in 8, 14 and 20 week-old (p ⟨0.05), but not 18 month-old aged mice. Histomorphometry showed that short-term SN reinstated load-responses in aged mice, with significant 33% and 117% increases in bone accrual at 47% and 37%, but not 27% of tibia length. Cortical responses to loading was heightened and widespread, now evident at all locations, following prolonged SN (108, 167 and 98% at 47, 37 and 27% of tibial length, respectively). In contrast, loading failed to modify trabecular bone mass or architecture. CONCLUSIONS: Mechanoadaptation become deficient with ageing and prolonging disuse amplifies this response in cortical but not trabecular bone.

Spontaneous osteoarthritis in Str/ort mice is unlikely due to greater vulnerability to mechanical trauma.

OBJECTIVE: Relative contributions of genetic and mechanical factors to osteoarthritis (OA) remain ill-defined. We have used a joint loading model found to produce focal articular cartilage (AC) lesions, to address whether genetic susceptibility to OA in Str/ort mice is related to AC vulnerability to mechanical trauma and whether joint loading influences spontaneous OA development. We also develop finite element (FE) models to examine whether AC thickness may explain any differential vulnerability to load-induced lesions. METHODS: Right knees of 8-week-old Str/ort mice were loaded, AC integrity scored and thickness compared to CBA mice. Mechanical forces engendered in this model and the impact of AC thickness were simulated in C57Bl/6 mice using quasi-static FE modelling. RESULTS: Unlike joints in non-OA prone CBA mice, Str/ort knees did not exhibit lateral femur (LF) lesions in response to applied loading; but exhibited thicker AC. FE modeling showed increased contact pressure and shear on the lateral femoral surface in loaded joints, and these diminished in joints containing thicker AC. Histological analysis of natural lesions in the tibia of Str/ort joints revealed that applied loading increased OA severity, proteoglycan loss and collagen type II degradation. CONCLUSION: Genetic OA susceptibility in Str/ort mice is not apparently related to greater AC vulnerability to trauma, but joint loading modifies severity of natural OA lesions in the medial tibia. FE modelling suggests that thicker AC in Str/ort mice diminishes tissue stresses and protects against load-induced AC lesions in the LF but that this is unrelated to their genetic susceptibility to OA.

Multimodal imaging demonstrates concomitant changes in bone and cartilage after destabilisation of the medial meniscus and increased joint laxity.

OBJECTIVE: Alterations in joint mechanics can cause osteoarthritis, which results in degeneration of both cartilage and bone tissue. The objective of this work is to measure changes in the laxity of the mouse knee joint after destabilisation of the medial meniscus (DMM) and to visualise and quantify the resulting three-dimensional changes in the bone and cartilage. METHODS: Skeletally mature C57Bl6 male mice underwent DMM surgery in the right leg. Animals were sacrificed immediately 0 weeks (n=15), 4 weeks (n=11) or 8 weeks (n=12) after surgery. For the 0-week group, the anterior-posterior (AP) and varus-valgus laxity of the DMM limb were compared to the contralateral limb. For 4 and 8-week groups, tibiae were scanned with micro-computed tomography (µCT) to quantify and visualise bone changes and with confocal scanning laser microscopy (CSLM) to measure changes in cartilage. RESULTS: Laxity testing measured an increase in AP range of motion, particularly in the anterior direction. The DMM limbs showed a decrease in epiphyseal trabecular bone at 8 weeks and a decrease in cartilage volume, primarily on the posterior medial plateau, compared to the contralateral limb. Significant bone remodelling was observed at the periphery of the joint and in severe cases, osteolysis extended through the growth plate. CONCLUSION: Multimodal imaging allowed quantifiable 3D assessment of bone and cartilage and indicated extensive changes in the tissues. The increase in AP laxity suggests that DMM surgery redistributes loading posteriorly on the medial plateau, resulting in bone and cartilage loss primarily on the posterior portion of the medial plateau.

Crack path selection in orientationally ordered composites.

While cracks in isotropic homogeneous materials propagate straight, perpendicularly to the tensile axis, cracks in natural and synthetic composites deflect from a straight path, often increasing the toughness of the material. Here we combine experiments and simulations to identify materials properties that predict whether cracks propagate straight or kink on a macroscale larger than the composite microstructure. Those properties include the anisotropy of the fracture energy, which we vary several fold by increasing the volume fraction of orientationally ordered alumina (Al_{2}O_{3}) platelets inside a polymer matrix, and a microstructure-dependent process zone size that is found to modulate the additional stabilizing or destabilizing effect of the nonsingular stress acting parallel to the crack. Those properties predict the existence of an anisotropy threshold for crack kinking and explain the surprisingly strong dependence of this threshold on sample geometry and load distribution.

Mechanobiological modeling of endochondral ossification: an experimental and computational analysis.

Long bone formation starts early during embryonic development through a process known as endochondral ossification. This is a highly regulated mechanism that involves several mechanical and biochemical factors. Because long bone development is an extremely complex process, it is unclear how biochemical regulation is affected when dynamic loads are applied, and also how the combination of mechanical and biochemical factors affect the shape acquired by the bone during early development. In this study, we develop a mechanobiological model combining: (1) a reaction-diffusion system to describe the biochemical process and (2) a poroelastic model to determine the stresses and fluid flow due to loading. We simulate endochondral ossification and the change in long bone shapes during embryonic stages. The mathematical model is based on a multiscale framework, which consisted in computing the evolution of the negative feedback loop between Ihh/PTHrP and the diffusion of VEGF molecule (on the order of days) and dynamic loading (on the order of seconds). We compare our morphological predictions with the femurs of embryonic mice. The results obtained from the model demonstrate that pattern formation of Ihh, PTHrP and VEGF predict the development of the main structures within long bones such as the primary ossification center, the bone collar, the growth fronts and the cartilaginous epiphysis. Additionally, our results suggest high load pressures and frequencies alter biochemical diffusion and cartilage formation. Our model incorporates the biochemical and mechanical stimuli and their interaction that influence endochondral ossification during embryonic growth. The mechanobiochemical framework allows us to probe the effects of molecular events and mechanical loading on development of bone.

Spatial relationship between bone formation and mechanical stimulus within cortical bone: Combining 3D fluorochrome mapping and poroelastic finite element modelling.

Bone is a dynamic tissue and adapts its architecture in response to biological and mechanical factors. Here we investigate how cortical bone formation is spatially controlled by the local mechanical environment in the murine tibia axial loading model (C57BL/6). We obtained 3D locations of new bone formation by performing 'slice and view' 3D fluorochrome mapping of the entire bone and compared these sites with the regions of high fluid velocity or strain energy density estimated using a finite element model, validated with ex-vivo bone surface strain map acquired ex-vivo using digital image correlation. For the comparison, 2D maps of the average bone formation and peak mechanical stimulus on the tibial endosteal and periosteal surface across the entire cortical surface were created. Results showed that bone formed on the periosteal and endosteal surface in regions of high fluid flow. Peak strain energy density predicted only the formation of bone periosteally. Understanding how the mechanical stimuli spatially relates with regions of cortical bone formation in response to loading will eventually guide loading regime therapies to maintain or restore bone mass in specific sites in skeletal pathologies.

Permeability and shear modulus of articular cartilage in growing mice.

Articular cartilage maturation is the postnatal development process that adapts joint surfaces to their site-specific biomechanical demands. Understanding the changes in mechanical tissues properties during growth is a critical step in advancing strategies for orthopedics and for cell- and biomaterial- based therapies dedicated to cartilage repair. We hypothesize that at the microscale, the articular cartilage tissue properties of the mouse (i.e., shear modulus and permeability) change with the growth and are dependent on location within the joint. We tested cartilage on the medial femoral condyle and lateral femoral condyle of seven C57Bl6 mice at different ages (2, 3, 5, 7, 9, 12, and 17 weeks old) using a micro-indentation test. Results indicated that permeability decreased with age from 2 to 17 weeks. Shear modulus reached a peak at the end of the growth (9 weeks). Within an age group, shear modulus was higher in the MFC than in the LFC, but permeability did not change. We have developed a method that can measure natural alterations in cartilage material properties in a murine joint, which will be useful in identifying changes in cartilage mechanics with degeneration, pathology, or treatment.

Predicting cortical bone adaptation to axial loading in the mouse tibia.

The development of predictive mathematical models can contribute to a deeper understanding of the specific stages of bone mechanobiology and the process by which bone adapts to mechanical forces. The objective of this work was to predict, with spatial accuracy, cortical bone adaptation to mechanical load, in order to better understand the mechanical cues that might be driving adaptation. The axial tibial loading model was used to trigger cortical bone adaptation in C57BL/6 mice and provide relevant biological and biomechanical information. A method for mapping cortical thickness in the mouse tibia diaphysis was developed, allowing for a thorough spatial description of where bone adaptation occurs. Poroelastic finite-element (FE) models were used to determine the structural response of the tibia upon axial loading and interstitial fluid velocity as the mechanical stimulus. FE models were coupled with mechanobiological governing equations, which accounted for non-static loads and assumed that bone responds instantly to local mechanical cues in an on-off manner. The presented formulation was able to simulate the areas of adaptation and accurately reproduce the distributions of cortical thickening observed in the experimental data with a statistically significant positive correlation (Kendall's τ rank coefficient τ = 0.51, p < 0.001). This work demonstrates that computational models can spatially predict cortical bone mechanoadaptation to a time variant stimulus. Such models could be used in the design of more efficient loading protocols and drug therapies that target the relevant physiological mechanisms.

Structure-mechanics relationships of collagen fibrils in the osteogenesis imperfecta mouse model.

The collagen molecule, which is the building block of collagen fibrils, is a triple helix of two α1(I) chains and one α2(I) chain. However, in the severe mouse model of osteogenesis imperfecta (OIM), deletion of the COL1A2 gene results in the substitution of the α2(I) chain by one α1(I) chain. As this substitution severely impairs the structure and mechanics of collagen-rich tissues at the tissue and organ level, the main aim of this study was to investigate how the structure and mechanics are altered in OIM collagen fibrils. Comparing results from atomic force microscopy imaging and cantilever-based nanoindentation on collagen fibrils from OIM and wild-type (WT) animals, we found a 33% lower indentation modulus in OIM when air-dried (bound water present) and an almost fivefold higher indentation modulus in OIM collagen fibrils when fully hydrated (bound and unbound water present) in phosphate-buffered saline solution (PBS) compared with WT collagen fibrils. These mechanical changes were accompanied by an impaired swelling upon hydration within PBS. Our experimental and atomistic simulation results show how the structure and mechanics are altered at the individual collagen fibril level as a result of collagen gene mutation in OIM. We envisage that the combination of experimental and modelling approaches could allow mechanical phenotyping at the collagen fibril level of virtually any alteration of collagen structure or chemistry.

Development of the femoral bicondylar angle in hominid bipedalism.

The bicondylar angle is the angle between the diaphysis of the femur and a line perpendicular to the infracondylar plane. The presence of a femoral bicondylar angle in Australopithecus afarensis indicates that these 3.5-million-year-old hominids were bipedal. Many studies have linked the formation of the femoral bicondylar angle with bipedality, but the mechanism for the formation of the angle is poorly understood. Mechanical factors, such as stresses and strains, influence the growth process. In particular, previous studies have demonstrated that hydrostatic compressive stress inhibits growth and ossification, and octahedral shear stress promotes growth and ossification. In this study we implemented these mechanobiological principles in a three-dimensional finite-element model of the distal femur. We applied loading conditions to the model to simulate loading during the single-leg stance phase of bipedal gait. The stresses in the physis of the distal femur that result from bipedal loading conditions promote growth and ossification more on the medial side than on the lateral side of the femur, forming the bicondylar angle. This model explains the presence of the bicondylar angle in hominid bipedalism and also the ontogenetic development of the bicondylar angle in growing children. The mechanobiological relationship between endochondral ossification and mechanical loading provides valuable insight into bone development and morphology.

Altered lacunar and vascular porosity in osteogenesis imperfecta mouse bone as revealed by synchrotron tomography contributes to bone fragility.

Osteogenesis imperfecta (brittle bone disease) is caused by mutations in the collagen genes and results in skeletal fragility. Changes in bone porosity at the tissue level indicate changes in bone metabolism and alter bone mechanical integrity. We investigated the cortical bone tissue porosity of a mouse model of the disease, oim, in comparison to a wild type (WT-C57BL/6), and examined the influence of canal architecture on bone mechanical performance. High-resolution 3D representations of the posterior tibial and the lateral humeral mid-diaphysis of the bones were acquired for both mouse groups using synchrotron radiation-based computed tomography at a nominal resolution of 700nm. Volumetric morphometric indices were determined for cortical bone, canal network and osteocyte lacunae. The influence of canal porosity architecture on bone mechanics was investigated using microarchitectural finite element (µFE) models of the cortical bone. Bright-field microscopy of stained sections was used to determine if canals were vascular. Although total cortical porosity was comparable between oim and WT bone, oim bone had more numerous and more branched canals (p<0.001), and more osteocyte lacunae per unit volume compared to WT (p<0.001). Lacunae in oim were more spherical in shape compared to the ellipsoidal WT lacunae (p<0.001). Histology revealed blood vessels in all WT and oim canals. µFE models of cortical bone revealed that small and branched canals, typical of oim bone, increase the risk of bone failure. These results portray a state of compromised bone quality in oim bone at the tissue level, which contributes to its deficient mechanical properties.

How tough is brittle bone? Investigating osteogenesis imperfecta in mouse bone.

The multiscale hierarchical structure of bone is naturally optimized to resist fractures. In osteogenesis imperfecta, or brittle bone disease, genetic mutations affect the quality and/or quantity of collagen, dramatically increasing bone fracture risk. Here we reveal how the collagen defect results in bone fragility in a mouse model of osteogenesis imperfecta (oim), which has homotrimeric α1(I) collagen. At the molecular level, we attribute the loss in toughness to a decrease in the stabilizing enzymatic cross-links and an increase in nonenzymatic cross-links, which may break prematurely, inhibiting plasticity. At the tissue level, high vascular canal density reduces the stable crack growth, and extensive woven bone limits the crack-deflection toughening during crack growth. This demonstrates how modifications at the bone molecular level have ramifications at larger length scales affecting the overall mechanical integrity of the bone; thus, treatment strategies have to address multiscale properties in order to regain bone toughness. In this regard, findings from the heterozygous oim bone, where defective as well as normal collagen are present, suggest that increasing the quantity of healthy collagen in these bones helps to recover toughness at the multiple length scales.

Characterising neovascularisation in fracture healing with laser Doppler and micro-CT scanning.

Vascularity of the soft tissues around a bone fracture is critical for successful healing, particularly when the vessels in the medullary canal are ruptured. The objective of this work was to use laser Doppler and micro-computer tomography (micro-CT) scanning to characterise neovascularisation of the soft tissues surrounding the fracture during healing. Thirty-two Sprague-Dawley rats underwent mid-shaft osteotomy of the left femur, stabilised with a custom-designed external fixator. Five animals were killed at each of 2, 4 days, 1, 2, 4 and 6 weeks post-operatively. Femoral blood perfusion in the fractured and intact contralateral limbs was measured using laser Doppler scanning pre- and post-operatively and throughout the healing period. At sacrifice, the common iliac artery was cannulated and infused with silicone contrast agent. Micro-CT scans of the femur and adjacent soft tissues revealed vessel characteristics and distribution in relation to the fracture zone. Blood perfusion dropped immediately after surgery and then recovered to greater than the pre-operative level by proliferation of small vessels around the fracture zone. Multi-modal imaging allowed both longitudinal functional and detailed structural analysis of the neovascularisation process.

What makes an accurate and reliable subject-specific finite element model? A case study of an elephant femur.

Finite element modelling is well entrenched in comparative vertebrate biomechanics as a tool to assess the mechanical design of skeletal structures and to better comprehend the complex interaction of their form-function relationships. But what makes a reliable subject-specific finite element model? To approach this question, we here present a set of convergence and sensitivity analyses and a validation study as an example, for finite element analysis (FEA) in general, of ways to ensure a reliable model. We detail how choices of element size, type and material properties in FEA influence the results of simulations. We also present an empirical model for estimating heterogeneous material properties throughout an elephant femur (but of broad applicability to FEA). We then use an ex vivo experimental validation test of a cadaveric femur to check our FEA results and find that the heterogeneous model matches the experimental results extremely well, and far better than the homogeneous model. We emphasize how considering heterogeneous material properties in FEA may be critical, so this should become standard practice in comparative FEA studies along with convergence analyses, consideration of element size, type and experimental validation. These steps may be required to obtain accurate models and derive reliable conclusions from them.