Application of a Fast Fourier Transform Method to the Characterization of the Elastic Behavior of Trabecular Bone

LUCAS COLABELLA; ADRIÁN CISILINO; PIOTR KOWALCZYK

Buenos Aires

Congreso; 1st Pan-American Congress on Computational Mechanics; 2015

International Center for Numerical Methods in Engineering

An alternative for using grafts or bone cement for the filling of bone cavities is the use of a bone scaffoldthat provides a temporary load-bearing function. Within the framework of the development of acomputational tool for the design of bone scaffolds, the application of a Fast Fourier Transform Method(FFTM) to the homogenization of the elastic behavior of trabecular bone is presented in this work.The FFTM, which was first introduced by Moulinec and Souquet [1], is based on the exact expression ofthe Green function for a linear elastic homogeneous material. The formulation of the method reduces theGreen function to the Lippmann-Schwinger integral equation, which is solved iteratively. The rate ofconvergence of the iterative procedure is directly related to the stiffness contrast between the phases. Thesimplicity of the FFTM greatly reduces the effort for the model data preparation when compared to FEM.The FFTM was implemented in C and programmed in parallel using OpenMP. The stress fields that resultfrom the FFTM are post processed to compute the macroscopic homogenized elastic tensor by means of theasymptotic homogenization method. This post processing is performed using a Matlab subroutine.The performance of the method is assessed for actual and artificial trabecular bone geometries. Thegeometries for the actual trabeculae are obtained from micro-tomography images. The images aresegmented and binarized to generate the models. The results for the homogenized elastic tensors arecompared to those obtained by Ibarra Pino and Cisilino [2], who also used the asymptotic homogenizationmethod, but based on finite-element stress results. The geometries of the artificial trabeculae are retrievedfrom the database developed by Kowalczyk [3]. These are computer-generated trabeculae with theirgeometries locally described by a set of scalar parameters. Their macroscopic (continuum) elasticproperties are assumed orthotropic and expressed as known functions of the geometric parameters.The FFT method is found suitable to provide results of the same accuracy of FEA. To this end, guidelinesare given for the model discretization and for the selection of the stiffness contrast between the void andbone phases. It is concluded that the FFT method can be effectively used for the homogenization of theelastic behavior of the trabecular structures with densities covering the range found in human bones.