A new mathematical model is presented for bone remodelling that includes a finite memory effect. In this new model, stimulus resulting from mechanical loading is separated from the signalling to grow or absorb bone. Also, a signal decaying exponentially to the distance from the point produced as well as an effect of decaying signal in time are considered. In addition, the model presented correctly predicts the synthesis and resorption of the bone tissue in a normal healthy mandible and in cases when teeth are missing. The model presented could be implemented to study cases of bone diseases where the signalling between the cells is disrupted and to predict changes in bone caused by several anomalies, such as missing a tooth or the presence of a dental prosthesis.

A previous bone remodelling model was presented elsewhere [30], and in the present paper, the same model was tested with new conditions; an interaction between bone tissue, bone substitute material and a dental implant was considered. The bone substitute material was assumed to be dead tissue, which does not synthesizes neither absorbs bone tissue, and it was considered, as well, resolvable. A moving border between the bone substitute material and the bone tissue was studied. The border moved as the newly synthesised bone tissue took over the bone substitute material. After the numerical calculations of time-steps, the whole bone substitute material was replaced by normal bone tissue and the implant was fixed in place only by bone tissue. Dynamical studies of the interaction of bone tissue and implant are used to improved implant design considering different factors, in this case, the presence of bone substitute material helping to fix the implant.