Rosy Paola Cárdenas Sandoval

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 on the cartilage canals growth

The role of cartilage canals is to transport nutrients and biological factors that cause the appearance of the secondary ossification centre (SOC). The SOC appears in the centre of the epiphysis of long bones. The canal development is a complex interaction between mechanical and biological factors that guide its expansion into the centre of the epiphysis. This article introduces the ‘Hypothesis on the growth of cartilage canals’. Here, we have considered that the development of these canals is an essential event for the appearance of SOC. Moreover, it is also considered to be important for the transport of molecular factors (RUNX2 and MMP9) at the ends of such canals. Once the canals are merged in the centre of the epiphysis, these factors are released causing hypertrophy of adjacent cells. This RUNX2 and MMP9 release occurs due to the action of mechanical loads that supports the epiphysis. In order to test this hypothesis, we use a hybrid approach using the finite element method to simulate the mechanical stresses present in the epiphysis and the cellular automata to simulate the expansion of the canals and the hypertrophy factors pathway. By using this hybrid approach, we have obtained as a result the spatial–temporal patterns for the growth of cartilage canals and hypertrophy factors within the epiphysis. The model is in qualitative agreement with experimental results previously reported by other authors. Thus, we conclude that this model may be used as a methodological basis to present a complete mathematical model of the processes involved in epiphyseal development.