The aim of this paper is to develop a multiscale hierarchical hybrid model based on finite element analysis and neural network computation to link mesoscopic scale (trabecular network level) and macroscopic (whole bone level) to simulate bone remodelling process. Because whole bone simulation considering the 3D trabecular level is time consuming, the finite element calculation is performed at macroscopic level and a trained neural network are employed as numerical devices for substituting the finite element code needed for the mesoscale prediction. The bone mechanical properties are updated at macroscopic scale depending on the morphological organization at the mesoscopic computed by the trained neural network. The digital image-based modeling technique using m-CT and voxel finite element mesh is used to capture 2 mm3 Representative Volume Elements at mesoscale level in a femur head. The input data for the artificial neural network are a set of bone material parameters, boundary conditions and the applied stress. The output data is the updated bone properties and some trabecular bone factors. The presented approach, to our knowledge, is the first model incorporating both FE analysis and neural network computation to simulate the multilevel bone adaptation in rapid way.
The bone adaptation process called remodeling process must address changes in its morphological and mechanical properties over time, at multiple levels, allowing for a more accurate description of the bone architecture [1][2]. This process occurs at different time and spatial scales in hierarchical way with interacting phenomenon between the different scales [1][2]. Therefore, an accurate simulation of bone remodeling should include different length scales. In the last few years, a general strategy has emerged for the design of multiscale methods in order to capture the macroscale behavior of the solutions. A large number of methods are based on numerical homogenization procedure [3], others introduce a new methods, and among them the artificial neural networks (NN) has attracted a growing interest in recent years [4]. In this paper, a hybrid method for bone remodeling multiscale simulation using finite element analysis and NN computation is proposed. The motivation to develop such hierarchical approach in the frame work of bone modeling is that a part of the mechanisms affecting bone strength are observed at trabecular level and cannot be described in precise way at macro level like microcracks accumulation, stress concentration du to tabecular network, bone cells activities and mass or chemical transport. Meshing the entire femur with its trabecular architecture generates some millions of finite elements with a huge computational time. The NN approach is beneficial if the numerical analysis of the complex model is time consuming or unfeasible [4].
The hybrid FENN method is a simulation procedure in which a continuum model is discretized into smaller submodels composed of RVE (fig. 1). Changes in the material distribution at macro level will have an effect on the stress/strain field, thus affecting the mechanical state of each RVE in the subsequent iteration. At the completion of every iteration, a new FE analysis is performed to update the mechanical parameters distribution in the macro level. A trained NN is applied locally to determine structural and mechanical change at meso level. The local results are passed back to the macro level.
The proposed methodology follows this iterative procedure until convergence is achieved.
A schematic illustration of the hierarchical approach is presented in Fig. 1.
The meso approach can be summarized by the five following steps: (i) FE remodeling simulation of the RVE for different combinations of bone inputs (ii) Averaging the RVE outputs.
Neural Networks models are composed of a large number of inter-connected processing elements called neurons, organized in layers. [8]. The single neuron performs a weighted sum of the inputs i
x that are generally the outputs of the neurons of the previous layer m v , adding threshold value i b and producing an output given by:
Where is a parameter defining the slope of the function.
To apply the NN, a training phase is needed which consists of an optimization procedure in order to determine the weights of the NN. the combinations of local applied stress (4 levels) and cycle frequency (5 levels) covering the range of mechanically observed/computed stresses and fequencies [5,6].
The set of equation describing the fully coupled change in the bone density is given by [5,6]: Where is the bone density, t is time and S is the coupled strain-damage stimulus function.
R , F and D denote respectively bone resorption rate, bone forming rate and damage resorption rate. R S , F S and D S denote respectively target levels of strain-damage energy density for bone resorption, formation and damage resorption.
The averaging relation of each RVE outputs, RVE i y , is expressed by :
Where o V and i y denote respectively the RVE reference domain and the output at every finite element location i.
To illustrate the capabilities of the FENN method, the remodeling of a 3D model of femur head has been studied. There are two materials in the macro model, trabecular bone and cortical bone. Marrow is considered only in the porous regions at the RVEs level. A 3D mesh is generated using tetrahedral elements. The model is run in alternating load and unload increments during 1000 iterations (days) with a fixed number of cycles per day (7000 cycles/day).
Since the investigation scale of the present work corresponds to one or some trabeculae level, we assume that the bone behaviour is purely elastic with isotropic averaged properties from nanoscale level.
In Figure 2 the distribution of the bone density is presented for both macro (a) and meso levels (b). The local RVE result is averaged and transferred to the entire femur for more accurate prediction of bone density.
In this study, the focus was on the development and the implementation of a novel multiscale approach for bone remodeling simulation using finite element simulation and neural network computation. The NN algorithm has been incorporated into a finite element code to link meso and macro s
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