It is well known that bone tissue adapts its shape and structure according to its mechanical environment. Bone adaptation occurs on the dense cortical bone and porous trabecular bone. The process of bone adaptation is shown to be dependent on a number of mechanical loading parameters such as magnitude, frequency, number of bouts etc. of applied loading through experimental studies. We propose to develop a numerical framework, which can simulate and predict cortical bone adaptation due to different parameters of loading. In pursuit of the development of the framework, we develop a method to generate finite element (FE) models of actual rat ulna from micro computed tomography (µ-CT) images. The external adaptation process is implemented in the model by moving the surface nodes of the FE mesh along the normal direction based on an evolution law characterized by two parameters:​ one that captures the rate of the adaptation process (referred to as gain);​ and the other characterizing the threshold value of the mechanical stimulus required for adaptation (referred to as threshold-sensitivity).

Cortical bone is firstly modeled as an elastic material. Loading from experiments of Robling et al [1] is applied on the FE model and the elastic boundary value problem is solved. Based on the results of the FE solution, the surface nodes are displaced according to the local strain energy density as the growth stimulus. Using this stimulus, we show that the model can simulate the effect of the magnitude of applied loading on the growth response. We calibrate the growth law parameters by comparing the results from our model to the experimental results. A parametric study is carried out to evaluate the effect of these two parameters on the adaptation response. We show, following comparison of results from the simulations to the experimental observations, that splitting the loading cycles into different number of bouts affects the threshold-sensitivity but not the rate of adaptation. We also show that the threshold-sensitivity parameter can quantify the mechanosensitivity of the osteocytes. The use of strain energy density stimulus and elastic material model cannot simulate the effect of frequency of applied loading on the cortical bone adaptation response. We model cortical bone as a poroelastic material to account for the interstitial fluid flow.

We aim to develop a growth stimulus similar to strain energy density for the poroelastic material model. In order to achieve this goal, we develop the FE model of a rectangular beam subjected to pure bending. This geometric model is chosen for simplicity, as an idealized representation of cortical bone. We then propose the use of the dissipation energy of the poroelastic flow as a mechanical stimulus for bone adaptation, and show that it can predict the effect of frequency of the applied load. Surface adaptation in the model depends on the weighted average of the mechanical stimulus in a “zone of influence” near each surface point, in order to incorporate the non-locality in the mechanotransduction of osteocytes present in the lacunae. We show that the dissipation energy stimulus and the resulting increase in second moment of inertia of the cross section increase linearly with frequency in the low frequency range (less than 10 Hz) and saturate at the higher frequency range (greater than 10 Hz). Similar non-linear adaptation frequency response also has been observed in numerous experiments. We extend the poroelastic material model, dissipation energy stimulus, and the zone of influence to the actual rat ulna FE model. We implement orthotropic permeability on the rat ulna model in order to be anatomically consistent. We calibrate the growth law parameters (gain and threshold-sensitivity) using experimental results. We analyze the growth response of cortical bone for a range of frequencies (from 2 Hz to 25 Hz) and show that the adaptation response is non-linear with respect to the frequency of loading