Micro–macro numerical modelling of bone regeneration in tissue engineering

Bone tissue regeneration using scaffolds inherently attends to two well-differentiated scales, i.e., the tissue level where the scaffold is implanted and the scaffold pore level where bone regeneration occurs. In this paper, this phenomenon is mathematically described and implemented as a two-length-scale coupled problem. Mechanical and fluid flow scaffold properties are macroscopically derived by means of the homogenization technique while the variables at the microscopic level are obtained by invoking the localization problem. As a first approach, cell migration within the scaffold is macroscopically modelled as a diffusion process based on the Fick’s law, with the diffusion coefficient depending on the size and spatial distribution of pores. At the microscopic scale, bone growth at the scaffold surface is considered to be proportional to the cell concentration and regulated by the local strain energy density. The mathematical model proposed has been numerically implemented using the finite element method (FEM) and the Voxel-FEM at the macroscopic and microscopic scales, respectively. The model has been qualitatively compared with experimental results found in the literature for a scaffold implanted in the femoral condyle of a rabbit achieving a good agreement.