Abstract
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 fur 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.