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

Abstract

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.

Flow-perfusion interferes with chondrogenic and hypertrophic matrix production by mesenchymal stem cells

“…Culturing chondrocytes under direct flow perfusion has been shown to stimulate proliferation and accumulation of cartilage markers (Beaupre et al, 2000;Pazzano et al, 2000;Freyria et al, 2005). However, fluid flow induces shear stress that may induce unwanted mechanical effects on chondrogenesis and terminal differentiation (Stevens et al, 1999;Carter and Wong, 2003;Garzon-Alvarado et al, 2011). A more thorough investigation of the effect of fluid flow on chondrogenic MSC differentiation is necessary to diminish potentially harmful effects of flow-perfusion in cartilage tissue engineering studies.…”

How do digits emerge? – mathematical models of limb development

“…While the core regulatory network, comprising PTHrP, IHH, and its receptor PTCH1, has been defined, it has remained unclear how the patterns and the spatio-temporal control of the process emerge from these interactions. A number of mathematical models have been developed to explain the distribution of the signalling proteins IHH and PTHrP and their impact on bone growth and development (Bougherara et al, 2010;Brouwers et al, 2006;Garzon-Alvarado et al, 2009;Garzon-Alvarado et al, 2010;Isaksson et al, 2008;van Donkelaar and Huiskes, 2007). Garzon-Alvarado and colleagues suggest that regulatory interactions between IHH and PTHrP result in Schnakenberg-like reaction kinetics (Garzon-Alvarado et al, 2009), which can give rise to Turing pattern (Gierer and Meinhardt, 1972).…”

The effect of electric fields on hyaline cartilage: an in vitro and in silico study

“…In this study, it was considered that endochondral ossification process is regulated by three molecules: Ihh, PTHrP and VEGF, the main regulators of bone growth [26], [28], [102], [114], [118], [120], [121].…”

“…For instance, the mathematical models implemented have simulated the effect of mechanical loading on the endochondral ossification, examining formation of the secondary center [113], [114], growth plate progression [115], and growth plate zone size [103], [104], and development of bone deformities [116]. Our group has previously used computational models based on the reaction-diffusion equations to explain how biochemical factors affect long bone formation, including the Ihh/PTHrP-negative loop and VEGF stimulator within the growth plate and bone rudiment [28], [105], [117], [118], the spongiosa primary development [104], [119], the growth pattern of the secondary ossification center [104], [117], [120], and the formation of cartilaginous canals in the epiphysis [121]. Previous mathematical models of endochondral ossification have studied the mechanical and biochemical factors involved in bone development separately.…”

“…Previous mathematical models of endochondral ossification have studied the mechanical and biochemical factors involved in bone development separately. Moreover, those models have evaluated long bone formation assuming a preexisting bone shape and ossification of the primary center [28], [103], [113], [120], [121]. Here we present a mathematical model that simulates the endochondral ossification process from a pure cartilage anlage to formation (but no ossification) of the epiphyses taking into account biochemical reactions and mechanical loading.…”