Efficient Harmonic Simulation of a Trabecular Bone Finite Element Model by means of Model Reduction

Three-dimensional serial reconstruction techniques allow us to develop very detailed microfinite element (micro-FE) model of bones that can very accurately represent the porous bone micro-architecture. However, such models are of very high dimension and, at present, simulation is limited to a linear elastic analysis only. In the present paper, we suggest to use model reduction in order to enable harmonic simulation for micro-FE models. We take two bone models of dimensions 130 000 and 900 000 and report results for implicit moment matching based via the Arnoldi process. We demonstrate that for the fist model a low-dimensional subspace of dimension 10 allows us to accurately describe frequency response up to 190 Hz. For the second model, a low-dimensional subspace of dimension 25 is enough to accurately describe frequency response up to 30 Hz. We show that the time to perform model reduction and then to simulate the low-dimensional model is orders of magnitude less than that needed for harmonic simulation of the original model. Introduction Fig. 1 sketches the micro finite element analysis [1]. Micro computer tomography (CT) is employed to make 3D high-resolution images (~50 microns) of a bone. Then the 3D reconstruction is directly transformed into an equally shaped micro finite element model by simply converting all bone voxels to equally sized 8-node brick elements. This results in finite element (FE) models with a very large number of elements (~ 1 10 million elements per cm) and special iterative solvers are usually required to solve such high dimensional problems. This allows us to simulate, for example, compression tests, and in this way it is possible to calculate the stiffness of a bone specimen while fully accounting for its micro-architecture. Such models can be used, for example to study differences in bone tissue loading between healthy and osteoporotic human bones during quasi static loading [2]. The main disadvantage of this approach is its huge computational requirements because of the high dimensionality of the models (on the order of 100 millions nodes). This led to the fact that such a computational analysis is limited to a static solution. There is increasing evidence, however, that bone responds in particular to dynamic loads [3]. It has been shown that the application of high-frequency, very low magnitude strains to a bone can prevent bone loss due to osteoporosis and can even result in increased bone strength in bones that are already osteoporotic. In order to better understand this phenomenon, it is necessary to determine the strain as sensed by the bone cells due to this loading. This would be possible with the micro-FE analysis, but then such an analysis need to be a dynamic one. Fig. 1. Micro finite element analysis. In the present study, it is investigated if a new technique called model order reduction can make the dynamic analysis feasible. Model order reduction is a relatively new area of mathematics that aims at finding a low-dimensional approximation for the high-dimensional system of ordinary differential equations. It has been used successfully over the last few years in different engineering disciplines like electrical engineering, structural mechanics, heat transfer, acoustics and electromagnetics [4]. We start by a short overview of how model order reduction can speed up harmonic simulation. Then we present results of harmonic analysis with the use of model reduction for the two microFE models representing bone specimens, one model with 42 528 and one with 305 066 nodes (number of equations is three times more). Harmonic Simulation by Means of Model Order Reduction After the discretization in space of a mechanical model, one obtains a system of ordinary differential equations of the second order as follows € M d x(t) dt 2 + E dx(t) dt + Kx(t) = Fu(t) (1)