Tomasz Lekszycki

In this paper, we present a novel theoretical model of bone evolution. The model takes into account growth of the blood vessels network as a real issue during the process. The stimulation of this growth is controlled by the signals from overloaded dying cells. A system of integro-differential equations describes changes in the density of blood vessels, bone cells and Young modulus, all of which define how the bone tissue evolves over time. The model considers several biomechanical signals such as changes in the density of elastic strain energy and nutrients. Two examples of changing bone tissue were examined to test the model: bone healing around a tooth implant and formation of osteophytes during osteoarthritis. In both cases, the effects of mechanical loading in the bone tissue evolution which are in agreement with the medical observations can be observed.

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.

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.

In the past years, many attempts have been made in order to model the process of bone remodeling. This process is complex, as it is governed by not yet completely understood biomechanical coupled phenomena. It is well known that bone tissue is able to self-adapt to different environmental demands of both mechanical and biological origin. The mechanical aspects are related to the functional purpose of the bone tissue, i.e., to provide support to the body and protection for the vitally important organs in response to the external loads. The many biological aspects include the process of oxygen and nutrients supply. To describe the biomechanical process of functional adaptation of bone tissue, the approach commonly adopted is to consider it as a ‘feedback’ control regulated by the bone cells, namely osteoblasts and osteoclasts. They are responsible for bone synthesis and resorption, respectively, while osteocytes are in charge of ‘sensing’ the mechanical status of the tissue. Within this framework, in Lekszycki and dell’Isola (ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik 92(6):426–444, 2012 ), a model based on a system of integro-differential equations was introduced aiming to predict the evolution of the process of remodeling in surgically reconstructed bones. The main idea in the aforementioned model was to introduce a scalar field, describing the biological stimulus regulating the interaction among all kinds of bone cells at a macroscale. This biological field was assumed to depend locally on certain deformation measures of the (reconstructed) bone tissue. However, biological knowledge suggests that this stimulus, after having been produced, ‘diffuses’ in bone tissue, so controlling in a complex way its remodeling. This means that the cells which are target of the stimulus may not be located in the same place occupied by the cells producing it. In this paper, we propose a model which intends to explain the diffusive nature of the biological stimulus to encompass the time-dependent and space–time displaced effects involved in bone reconstruction process. Preliminary numerical simulations performed in typical cases are presented. These numerical case studies suggest that the ‘diffusive’ model of stimulus is promising: we plan to continue these kinds of studies in further investigations.