A mathematical model of epiphyseal development: hypothesis of growth pattern of the secondary ossification centre

This paper introduces a 'hypothesis about the growth pattern of the secondary ossification centre (SOC)', whereby two phases are assumed. First, the formation of cartilage canals as an event essential for the development of the SOC. Second, once the canals are merged in the central zone of the epiphysis, molecular factors are released (primarily Runx2 and MMP9) spreading and causing hypertrophy of adjacent cells. In addition, there are two important molecular factors in the epiphysis: PTHrP and Ihh. The first one inhibits chondrocyte hypertrophy and the second helps the cell proliferation. Between these factors, there is negative feedback, which generates a highly localised and stable pattern over time. From a mathematical point of view, this pattern is similar to the patterns of Turing. The spread of Runx2 hypertrophies the cells from the centre to the periphery of the epiphysis until found with high levels of PTHrP to inhibit hypertrophy. This mechanism produces the epiphyseal bone-plate. Moreover, the hypertrophy is inhibited when the cells sense low shear stress and high pressure levels that maintain the articular cartilage structure. To test this hypothesis, we solve a system of coupled partial differential equations using the finite element method and we have obtained spatio-temporal patterns of the growth process of the SOC. The model is in qualitative agreement with experimental results previously reported by other authors. Thus, we conclude that this model can be used as a methodological basis to present a complete mathematical model of the whole epiphyseal development.     

A mathematical model of epiphyseal development: hypothesis of growth pattern of the secondary ossification centre

This paper introduces a ‘hypothesis about the growth pattern of the secondary ossification centre (SOC)’, whereby two phases are assumed. First, the formation of cartilage canals as an event essential for the development of the SOC. Second, once the canals are merged in the central zone of the epiphysis, molecular factors are released (primarily Runx2 and MMP9) spreading and causing hypertrophy of adjacent cells. In addition, there are two important molecular factors in the epiphysis: PTHrP and Ihh. The first one inhibits chondrocyte hypertrophy and the second helps the cell proliferation. Between these factors, there is negative feedback, which generates a highly localised and stable pattern over time. From a mathematical point of view, this pattern is similar to the patterns of Turing. The spread of Runx2 hypertrophies the cells from the centre to the periphery of the epiphysis until found with high levels of PTHrP to inhibit hypertrophy. This mechanism produces the epiphyseal bone-plate. Moreover, the hypertrophy is inhibited when the cells sense low shear stress and high pressure levels that maintain the articular cartilage structure. To test this hypothesis, we solve a system of coupled partial differential equations using the finite element method and we have obtained spatio-temporal patterns of the growth process of the SOC. The model is in qualitative agreement with experimental results previously reported by other authors. Thus, we conclude that this model can be used as a methodological basis to present a complete mathematical model of the whole epiphyseal development.     

Epidermal Growth Factor Receptor (EGFR) Signaling Regulates Epiphyseal Cartilage Development through β-Catenin-dependent and -independent Pathways

The epidermal growth factor receptor (EGFR) is an essential player in the development of multiple organs during embryonic and postnatal stages. To understand its role in epiphyseal cartilage development, we generated transgenic mice with conditionally inactivated EGFR in chondrocytes. Postnatally, these mice exhibited a normal initiation of cartilage canals at the perichondrium, but the excavation of these canals into the cartilage was strongly suppressed, resulting in a delay in the formation of the secondary ossification center (SOC). This delay was accompanied by normal chondrocyte hypertrophy but decreased mineralization and apoptosis of hypertrophic chondrocytes and reduced osteoclast number at the border of marrow space. Immunohistochemical analyses demonstrated that inactivation of chondrocyte-specific EGFR signaling reduced the amounts of matrix metalloproteinases (MMP9, -13, and -14) and RANKL (receptor activator of NF-κB ligand) in the hypertrophic chondrocytes close to the marrow space and decreased the cartilage matrix degradation in the SOC. Analyses of EGFR downstream signaling pathways in primary epiphyseal chondrocytes revealed that up-regulation of MMP9 and RANKL by EGFR signaling was partially mediated by the canonical Wnt/β-catenin pathway, whereas EGFR-enhanced MMP13 expression was not. Further biochemical studies suggested that EGFR signaling stimulates the phosphorylation of LRP6, increases active β-catenin level, and induces its nuclear translocation. In line with these in vitro studies, deficiency in chondrocyte-specific EGFR activity reduced β-catenin amount in hypertrophic chondrocytes in vivo. In conclusion, our work demonstrates that chondrocyte-specific EGFR signaling is an important regulator of cartilage matrix degradation during SOC formation and epiphyseal cartilage development and that its actions are partially mediated by activating the β-catenin pathway.     

Development of the cartilage canals and the secondary center of ossification in the distal chondroepiphysis of the prenatal human femur.

The cartilaginous epiphysis of the distal femur is vascularized by a network of cartilage canals during prenatal development. The vascular invasion of the epiphysis begins at approximately eight to ten weeks of gestation with the initiation of cartilage canal formation. A complex vascular system develops within the canals and is well defined by fourteen weeks of gestation. The vascular system is fully developed several months prior to the development of the secondary center of ossification. The formation of the secondary center of ossification within the distal femoral epiphysis is preceded by changes that occur simultaneously within both the chondrocytes in the central portion of the epiphysis and the vascular and perivascular elements contained within the cartilage canals in the central portion of the epiphysis. These concurrent changes in the cellular morphology of the central chondrocytes and in the cellular structure of the central cartilage canals appear to be linked with the initiation of the process of osteogenesis.     

Role of tartrate-resistant acid phosphatase (TRAP) in long bone development

Tartrate resistant acid phosphatase (TRAP) was shown to be critical for skeleton development, and TRAP deficiency leads to a reduced resorptive activity during endochondral ossification resulting in an osteopetrotic phenotype and shortened long bones in adult mice. A proper longitudinal growth depends on a timely, well-coordinated vascularization and formation of the secondary ossification center (SOC) of the long bones epiphysis. Our results demonstrate that TRAP is not essential for the formation of the epiphyseal vascular network. Therefore, in wild type (Wt) controls as well as TRAP deficient (TRAP(-/-)) mutants vascularised cartilage canals are present from postnatal day (P) five. However, in the epiphysis of the TRAP(-/-) mice cartilage mineralization, formation of the marrow cavity and the SOC occur prematurely compared with the controls. In the mutant mice the entire growth plate is widened due to an expansion of the hypertrophic zone. This is not seen in younger animals but first detected at week (W) three and during further development. Moreover, an enhanced number of thickened trabeculae, indicative of the osteopetrotic phenotype, are observed in the metaphysis beginning with W three. Epiphyseal excavation was proposed as an important function of TRAP, and we examined whether TRAP deficiency affects this process. We therefore evaluated the marrow cavity volume (MCV) and the epiphyseal volume (EV) and computed the MCV to EV ratio (MCV/EV). We investigated developmental stages until W 12. Our results indicate that both epiphyseal excavation and establishment of the SOC are hardly impaired in the knockouts. Furthermore, no differences in the morphology of the epiphyseal bone trabeculae and remodeling of the articular cartilage layers are noted between Wt and TRAP(-/-) mice. We conclude that in long bones, TRAP is critical for the development of the growth plate and the metaphysis but apparently not for the epiphyseal vascularization, excavation, and establishment of the SOC.     

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.     

Chondrocytes are released as viable cells during cartilage resorption associated with the formation of intrachondral canals in the rat tibial epiphysis

The development of cartilage canals is the first event of the ossification of the epiphyses in mammals. Canal formation differs from vascular invasion during primary ossification, since the former involves resorption of resting cartilage and is uncoupled from bone deposition. To learn more about the fate of resorbed chondrocytes during this process, we have carried out structural, cell proliferation, and in situ hybridization studies during the first stages of ossification of the rat tibial proximal epiphysis. Results concerning the formation of the cartilage canals implied the release of resting chondrocytes from the cartilage matrix to the canal cavity. Released chondrocytes had a well-preserved structure, expressed type-II collagen, and maintained the capacity to divide. All these data suggested that chondrocytes released into the canals remained viable for a specific time. Analysis of the proliferative activity at different regions of the cartilage canals showed that the percentage of proliferative chondrocytes at areas of active cartilage resorption was significantly higher than that in zones of low resorption. These results are consistent with the hypothesis that resting chondrocytes surrounding canals have a role in supplying cells for the development of the secondary ossification center. Since released chondrocytes are at an early stage of differentiation greatly preceding their entry into the apoptotic pathway and are exposed to a specific matrix, cellular, and humoral microenvironment, they might differentiate to other cell types and contribute to the ossification of the epiphysis.This research was supported by the Ministerio de Ciencia y Tecnología (Spain), grant no. MCT-00-BMC-0446. The Instituto Universitario de Oncología is financed by Obra Social Cajastur-Asturias, Spain. J. Alvarez receives financial support from the Ministerio de Ciencia y Tecnología (CAJAL-03-06) and L. Costales from the Ministerio de Ciencia y Tecnología (MCYT, FP2000-5486).     

The distribution of type VI collagen in the developing tissues of the bovine femoral head

Type VI collagen appears central to the maintenance of tissue integrity. In adult articular cartilage, type VI collagen is preferentially localised in the chondron where it may be involved in cell attachment. In actively remodelling developing cartilage, the distribution is less certain. We have used confocal immunohistochemistry and in situ hybridisation to investigate type VI collagen distribution in third trimester bovine proximal femoral epiphyses. In general, type VI collagen immunofluorescence was concentrated in the chondrocyte pericellular matrix, with staining intensity strongest in regions which persist to maturity and weakest in regions that remodel during development. Type VI collagen was also present in cartilage canals. In the growth plate and around the secondary centre of ossification, the intensity of type VI collagen stain rapidly decreased with chondrocyte maturation and was absent at hypertrophy, except where canal branches penetrated the growth plate and stain was retained around the adjacent chondrocytes. In situ hybridisation confirmed the presence of type VI collagen mRNA in cartilage canal mesenchymal cells but the signal was low in chondrocytes, suggesting minimal levels of synthesis and turnover. The results are consistent with a role for type VI collagen in stabilising the extracellular matrix during development.     

Gene expression and distribution of connective tissue growth factor (CCN2/CTGF) during secondary ossification center formation.

CCN2/connective tissue growth factor (CCN2/CTGF) is a critical signaling modulator of mesenchymal tissue development. This study investigated the localization and expression of CCN2/CTGF as a factor supporting angiogenesis and chondrogenesis during development of secondary ossification centers in the mouse tibial epiphysis. Formation of the secondary ossification center was initiated by cartilage canal formation and blood vessel invasion at 7 days of age, and onset of ossification was observed at 14 days. In situ hybridization showed that CCN2/CTGF mRNA was distinctively expressed in the region of the cartilage canal and capsule-attached marginal tissues at 7 days of age, and distinct expression was also observed in proliferating chondrocytes around the marrow space at 14 days of age. Immunostaining showed that CCN2/CTGF was distributed broadly around the expressed cells located in the central region of the epiphysis, where the chondrocytes become hypertrophic and the cartilage canal enters into the hypertrophic mass. Furthermore, an overlapping distribution of metalloproteinase (MMP)9 and CCN2/CTGF was found in the secondary ossification center. These findings suggest that the CCN2/CTGF is involved in establishing epiphyseal vascularization and remodeling, which eventually determines the secondary ossification center in the developing epiphysial cartilage.     

Retrograde tracing and neuropeptide immunohistochemistry of sensory neurones projecting to the cartilaginous distal femoral epiphysis of young rats

Although cartilage is considered to be devoid of innervation, axons occur in the perichondrium and during development in cartilage canals, thereby having a relatively close spatial relationship to chondroblasts and chondrocytes. The present study locates the source of the sensory innervation of the femoral cartilaginous epiphyses of young rats and investigates whether the neuropeptide calcitonin gene-related peptide (CGRP) can influence chondrocytes. Retrograde tracing from the distal femoral epiphysis of young rats with Fast Blue (FB) showed labelled neuronal profiles in the L2-L5 dorsal root ganglia. Sample countings indicated that 50% of the FB-labelled neuronal profiles were located at the L3 level and 25% at the L4 level. The labelled neurones had diameters of 15-40 microm, with a peak at 25-30 microm. Immunohistochemistry showed that about 50% of the FB-labelled profiles contained CGRP. Together with the finding that CGRP influences bone cells to generate the second messenger cAMP, this result suggested the hypothesis that chondrocytes might be similarly influenced by CGRP. However, stimulation of cartilage slices with CGRP in vitro followed by an assay of the cAMP content did not provide support for this hypothesis. We conclude that primary sensory neurones containing CGRP project to the perichondrium and to cartilage canals of growing cartilage, and that exogenous CGRP does not elevate the cAMP content of cartilage slices in vitro.     

Can the size of the epiphysis determine the number of secondary ossification centers? A mathematical approach

Previous studies have concluded that the chemical feedback (loop) between two reagent molecular factors through a reaction-diffusion mechanism could explain the stable spatial pattern found in the origin of the secondary ossification centres (SOCs). Furthermore, the emergence of the SOC may depend on the size and shape of the head of the bone, as observed in different animals. In this paper, we develop new computer simulations that study the effect of the size of the epiphysis on the emergence of the SOC. This study determines two or more SOCs, that may appear in the head of long bones, depending on the size of the epiphysis.     

Can the size of the epiphysis determine the number of secondary ossification centers? A mathematical approach

Previous studies have concluded that the chemical feedback (loop) between two reagent molecular factors through a reaction–diffusion mechanism could explain the stable spatial pattern found in the origin of the secondary ossification centres (SOCs). Furthermore, the emergence of the SOC may depend on the size and shape of the head of the bone, as observed in different animals. In this paper, we develop new computer simulations that study the effect of the size of the epiphysis on the emergence of the SOC. This study determines two or more SOCs, that may appear in the head of long bones, depending on the size of the epiphysis.     

A fully coupled poroelastic reactive-transport model of cartilage

Cartilage maintains its integrity in a hostile mechanical environment. This task is made more difficult because cartilage has no blood supply, and so nutrients and growth factors need to be transported greater distances than normal to reach cells several millimetres from the cartilage surface. The chondrocytes embedded within the extracellular matrix (ECM) are essential for maintaining the mechanical integrity of the ECM, through a balance of degradation and synthesis of collagen and proteoglycans. A chondrocyte senses various chemical and mechanical signals in its local microenvironment, responding by appropriate adaption of the local ECM. Clearly a 'systems understanding' of cartilage behaviour is of critical importance in developing an integrated understanding of both normal and abnormal physiology of cartilage. In a series of papers, we have developed a reactive-transport porous-media model to investigate the coupled processes of growth factor transport, mechanical deformation and fluid flow, and in this paper, we extend the model to include biosynthesis and degradation of matrix molecules. The model is validated using three independent experimental data sets, it being found that a single set of parameters described the experimental results remarkably well. The model is then employed to make predictions about changes in proteoglycan content under a variety of conditions. This model may prove useful in predicting the behaviour of tissue engineering constructs, or predicting the outcome of repair processes in cartilage.     

Parathyroid hormone-related peptide-depleted mice show abnormal epiphyseal cartilage development and altered endochondral bone formation

To elucidate the role of PTHrP in skeletal development, we examined the proximal tibial epiphysis and metaphysis of wild-type (PTHrP-normal) 18- 19-d-old fetal mice and of chondrodystrophic litter mates homozygous for a disrupted PTHrP allele generated via homologous recombination in embryonic stem cells (PTHrP-depleted). In the PTHrP-normal epiphysis, immunocytochemistry showed PTHrP to be localized in chondrocytes within the resting zone and at the junction between proliferative and hypertrophic zones. In PTHrP-depleted epiphyses, a diminished [3H]thymidine-labeling index was observed in the resting and proliferative zones accounting for reduced numbers of epiphyseal chondrocytes and for a thinner epiphyseal plate. In the mutant hypertrophic zone, enlarged chondrocytes were interspersed with clusters of cells that did not hypertrophy, but resembled resting or proliferative chondrocytes. Although the overall content of type II collagen in the epiphyseal plate was diminished, the lacunae of these non-hypertrophic chondrocytes did react for type II collagen. Moreover, cell membrane-associated chondroitin sulfate immunoreactivity was evident on these cells. Despite the presence of alkaline phosphatase activity on these nonhypertrophic chondrocytes, the adjacent cartilage matrix did not calcify and their persistence accounted for distorted chondrocyte columns and sporadic distribution of calcified cartilage. Consequently, in the metaphysis, bone deposited on the irregular and sparse scaffold of calcified cartilage and resulted in mixed spicules that did not parallel the longitudinal axis of the tibia and were, therefore, inappropriate for bone elongation. Thus, PTHrP appears to modulate both the proliferation and differentiation of chondrocytes and its absence alters the temporal and spatial sequence of epiphyseal cartilage development and of subsequent endochondral bone formation necessary for normal elongation of long bones.     

Mechanics of chondrocyte hypertrophy

Chondrocyte hypertrophy is a characteristic of osteoarthritis and dominates bone growth. Intra- and extracellular changes that are known to be induced by metabolically active hypertrophic chondrocytes are known to contribute to hypertrophy. However, it is unknown to which extent these mechanical conditions together can be held responsible for the total magnitude of hypertrophy. The present paper aims to provide a quantitative, mechanically sound answer to that question. To address this aim requires a quantitative tool that captures the mechanical effects of collagen and proteoglycans, allows temporal changes in tissue composition, and can compute cell and tissue deformations. These requirements are met in our numerical model that is validated for articular cartilage mechanics, which we apply to quantitatively explain a range of experimental observations related to hypertrophy. After validating the numerical approach for studying hypertrophy, the model is applied to evaluate the direct mechanical effects of axial tension and compression on hypertrophy (Hueter-Volkmann principle) and to explore why hypertrophy is reduced in case of partially or fully compromised proteoglycan expression. Finally, a mechanical explanation is provided for the observation that chondrocytes do not hypertrophy when enzymatical collagen degradation is prohibited (S1Pcko knock-out mouse model). This paper shows that matrix turnover by metabolically active chondrocytes, together with externally applied mechanical conditions, can explain quantitatively the volumetric change of chondrocytes during hypertrophy. It provides a mechanistic explanation for the observation that collagen degradation results in chondrocyte hypertrophy, both under physiological and pathological conditions.     

A computational model to explore the role of angiogenic impairment on endochondral ossification during fracture healing

While it is well established that an adequate blood supply is critical to successful bone regeneration, it remains poorly understood how progenitor cell fate is affected by the altered conditions present in fractures with disrupted vasculature. In this study, computational models were used to explore how angiogenic impairment impacts oxygen availability within a fracture callus and hence regulates mesenchymal stem cell (MSC) differentiation and bone regeneration. Tissue differentiation was predicted using a previously developed algorithm which assumed that MSC fate is governed by oxygen tension and substrate stiffness. This model was updated based on the hypothesis that cell death, chondrocyte hypertrophy and endochondral ossification are regulated by oxygen availability. To test this, the updated model was used to simulate the time course of normal fracture healing, where it successfully predicted the observed quantity and spatial distribution of bone and cartilage at 10 and 20 days post-fracture (dpf). It also predicted the ratio of cartilage which had become hypertrophic at 10 dpf. Following this, three models of fracture healing with increasing levels of angiogenic impairment were developed. Under mild impairment, the model predicted experimentally observed reductions in hypertrophic cartilage at 10 dpf as well as the persistence of cartilage at 20 dpf. Models of more severe impairment predicted apoptosis and the development of fibrous tissue. These results provide insight into how factors specific to an ischemic callus regulate tissue regeneration and provide support for the hypothesis that chondrocyte hypertrophy and endochondral ossification during tissue regeneration are inhibited by low oxygen.     

EGR1 promotes the cartilage degeneration and hypertrophy by activating the Krüppel-like factor 5 and β-catenin signaling

Osteoarthritis is one of the most common orthopedic diseases in elderly people who have lost their mobility. In this study,we observed abnormally high EGR1 expression in the articular cartilage of patients with osteoarthritis. We also found significantly high EGR1 expression in the articular cartilage of mice with destabilized medial meniscus (DMM)-induced osteoarthritis and 20-month-old mice. In vitro experiments indicated that IL-1β could significantly enhance EGR1 expression in primary mouse chondrocytes. EGR1 over-expression in chondrocytes using adenovirus could inhibit COl2A1 expression and enhance MMP9 and MMP13 expression. And silencing EGR1, using RNAi, had the opposite effects. Moreover, EGR1 over-expression accelerated chondrocyte hypertrophy in vitro, and EGR1 knockdown reversed this effect. We then explored the underlying mechanism. EGR1 over-expression increased Kruppel-Like Factor 5 (KLF5) protein level without influencing its synthesis. Enhanced EGR1 expression induced its integration with KLF5, leading to suppressed ubiquitination of KLF5. Moreover, EGR1 prompted β-catenin nuclear transportation to control chondrocyte hypertrophy. Ectopic expression of EGR1 in articular cartilage aggravated the degradation of the cartilage matrix in vivo. The EGR1 inhibitor, ML264, protected chondrocytes from IL-1β-mediated cartilage matrix degradation in vitro and DMM-induced osteoarthritis in vivo. Above all, we demonstrate the effect and mechanisms of EGR1 on osteoarthritis and provide evidence that the ML264 might be a potential drug for treating osteoarthritis in the future.     

Hypoxia Inhibits Hypertrophic Differentiation and Endochondral Ossification in Explanted Tibiae

Purpose

Hypertrophic differentiation of growth plate chondrocytes induces angiogenesis which alleviates hypoxia normally present in cartilage. In the current study, we aim to determine whether alleviation of hypoxia is merely a downstream effect of hypertrophic differentiation as previously described or whether alleviation of hypoxia and consequent changes in oxygen tension mediated signaling events also plays an active role in regulating the hypertrophic differentiation process itself.

Materials and Methods

Fetal mouse tibiae (E17.5) explants were cultured up to 21 days under normoxic or hypoxic conditions (21% and 2.5% oxygen respectively). Tibiae were analyzed on growth kinetics, histology, gene expression and protein secretion.

Results

The oxygen level had a strong influence on the development of explanted fetal tibiae. Compared to hypoxia, normoxia increased the length of the tibiae, length of the hypertrophic zone, calcification of the cartilage and mRNA levels of hypertrophic differentiation-related genes e.g. MMP9, MMP13, RUNX2, COL10A1 and ALPL. Compared to normoxia, hypoxia increased the size of the cartilaginous epiphysis, length of the resting zone, calcification of the bone and mRNA levels of hyaline cartilage-related genes e.g. ACAN, COL2A1 and SOX9. Additionally, hypoxia enhanced the mRNA and protein expression of the secreted articular cartilage markers GREM1, FRZB and DKK1, which are able to inhibit hypertrophic differentiation.

Conclusions

Collectively our data suggests that oxygen levels play an active role in the regulation of hypertrophic differentiation of hyaline chondrocytes. Normoxia stimulates hypertrophic differentiation evidenced by the expression of hypertrophic differentiation related genes. In contrast, hypoxia suppresses hypertrophic differentiation of chondrocytes, which might be at least partially explained by the induction of GREM1, FRZB and DKK1 expression.