A new approach for initial callus growth during fracture healing in long bones

• New approach for initial callus growth in bone fracture healing. • Finite Element approach of the diffusion problem concerning biological magnitudes. • Geometrical algorithm for callus growth with automatic remeshing. • Natural callus growth, without previous mesh affecting conditions and direction of growth. The incidence of bone fracture has become a major clinical problem on a worldwide scale. In the past two decades there has been an increase in the use of computational tools to analyse the bone fracture problem. In several works, various study cases have been analysed to compare human and animal bone fracture healing. Unfortunately, there are not many publications about computational advances in this field and the existing approaches to the problem are usually similar. In this context, the objective of this work is the application of a diffusion problem in the model of the bone fragments resulting from fracture, working together with a mesh-growing algorithm that allows free growth of the callus depending on the established conditions, without a pre-meshed domain. The diffusion problem concerns the different biological magnitudes controlling the callus growth, among which Mesenchymal Stem Cells and chondrocytes concentrations were chosen, together with Tumour Necrosis Factor α and Bone Morphogenetic Protein as the factors influencing the velocity in the callus formation. A Finite Element approach was used to solve the corresponding diffusion problems, obtaining the concentration values along the entire domain and allowing detecting the zones in which biological magnitudes reach the necessary thresholds for callus growth. The callus growth is guided by a geometrical algorithm which performs an additional mesh generation process (self-added mesh) at each step of the iterative procedure until complete callus formation. The proposed approach was applied to different types of diaphyseal femoral fractures treated by means of intramedullary nailing. Axisymmetric models based on triangular quadratic elements were used, obtaining results in good agreement with clinical evidence of these kinds of fractures. The algorithm proposed has the advantage of a natural callus growth, without the existence of a previous mesh that may affect the conditions and direction of growth. The approach is intended for the initial phase of callus growth. Future work will address the implementation of the corresponding formulations for tissue transformation and bone remodelling in order to achieve complete fracture healing.