A mathematical model of epiphyseal development: hypothesis on the cartilage canals growth

Abstract

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.

Spongiosa Primary Development: A Biochemical Hypothesis by Turing Patterns Formations

“…The cartilage tissue molds are formed through the condensation of mesenchymal cells [5] followed by their differentiation into chondrocytes (cells that produce and maintain cartilage matrix) and secretion of typical components of the extracellular matrix of cartilage [6]. Once the mold of cartilage is formed, it is invaded initially in its center and then at each end by a mixture of cells that give rise to the appearance of primary and secondary centers of ossification, respectively, [7–9]. The ossification centers invade the cartilage gradually until it is completely replaced by bone tissue, except the articular surfaces.…”

“…The process of endochondral ossification has been studied for several years, and different models have been developed in silico , verified by histological reports, and in vivo experiments, which have tried to explain the process of bone formation through this mechanism [7–9, 13, 14, 18–21]. For example, Courtin et al [18] in their work performed the comparison between the sequence of morphological events involved in embryonic bone formation and spatiotemporal characteristics of self-organization generated by a reaction-diffusion model related to the metabolism of the periosteal bone mineralization.…”

“…The hypothesis of Courtin et al is based on the role of self-organization of mineralization of bone metabolism, which gives rise to a well-organized space architecture. Subsequent research such as those by Garzón-Alvarado et al [7–9, 22] have raised different hypotheses about the interaction of mechanical, cellular, and molecular factors that lead to the formation of secondary ossification centers in the epiphyses of long bones, and it also helped the development and bone growth and the primary bone formation. These hypotheses suggest that biological processes and interactions between different factors can be represented using mathematical models, where the chemical feedback among molecular reagent factors through reaction-diffusion mechanisms may explain the stable spatial patterns found in the origin of the secondary ossification centers and in the formation of cartilage canals.…”

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

“…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.…”