A stochastic micro to macro mechanical model for the evolution of bone-implant interface stiffness

Upon placement of an implant into living bone, an interface is formed through which various biochemical, biological, physical, and mechanical interactions take place. This interface evolves over time as the mechanical properties of peri-implant bone increase. Owing to the multifactorial nature of interfacial processes, it is challenging to devise a comprehensive model for predicting the mechanical behavior of the bone-implant interface. We propose a simple spatio-temporally evolving mechanical model - from an elementary unit cell comprising randomly oriented mineralized collagen fibrils having randomly assigned stiffness all the way up to a macroscopic bone-implant interface in a gap healing scenario. Each unit cell has an assigned Young's modulus value between 1.62 GPa and 25.73 GPa corresponding to minimum (i.e., 0) and maximum (i.e., 0.4) limits of mineral volume fraction, respectively, in the overlap region of the mineralized collagen fibril. Gap closure and subsequent stiffening are modeled to reflect the two main directions of peri-implant bone formation, i.e., contact osteogenesis and distance osteogenesis. The linear elastic stochastic finite element model reveals highly nonlinear temporal evolution of bone-implant interface stiffness, strongly dictated by the specific kinetics of contact osteogenesis and distance osteogenesis. The bone-implant interface possesses a small stiffness until gap closure, which subsequently evolves into a much higher stiffness, and this transition is reminiscent of a percolation transition whose threshold corresponds to gap closure. The model presented here, albeit preliminary, can be incorporated into future calculations of the bone-implant system where the interface is well-defined mechanically. STATEMENT OF SIGNIFICANCE: A simple, physically informed model for the mechanical characteristics of the bone-implant interface is still missing. Here, we start by extending the reported mechanical characteristics of a one cubic micrometre unit cell to a 250 µm long interface made of 1 µm thick layers. The stiffness of each cell (based on mineral content) is assigned randomly to mimic bone micro-heterogeneity. The numerical study of this interface representative structure allows for the simultaneous determination of the spatio-temporal evolution of the mechanical response at local (discrete element) and global (overall model) scales. The proposed model is the first of this kind that can easily be incorporated into realistic future models of bone-implant interaction with emphasis on implant stability and different loading conditions.