Keywords: image-based CAE, voxel, element by element, bone microstructure, remodeling, parallel computing, PC-cluster.

The Image-Based approach enables modeling of natural components such as living tissue, directly from the X-ray micro CT (Computer Tomography) scanning and the biomechanics field of bone benefits from this approach and finite element analysis that could predict the mechanical environment of complicated bone microstructure [1,2]. The image based finite element model can be constructed by piling up the pictures obtained from micro-CT scanner with enough resolution to represent the trabecular structures. Because the morphological changes in the trabeculae are regarded as a local stress regulation process due to remodeling activities, Adachi et al. [3,4] have been simulated trabecular structural adaptation that depends on non-uniformity of stress and strain distribution as a measure of the local mechanical stimulus of remodeling. In image-based approach, the adaptation is realized by simple removal and addition of voxel elements on the surface of bone microstructure.

Recently, more computing power is required for accurate and detailed analysis, such as computational simulation for entire bone with more detailed microstructure with high resolution. The most time consuming process of the large scale finite element analysis is the solution process of the large linear equations. Because the image-based model of bone microstructure is consisted of uniform voxel elements, the element by element method [5,6] is employed as solution technique for conjugate gradient method.

In this paper, efficient computation approach to solve large scale remodeling procedure of bone microstructure with parallel computing technology is presented. The diagonal preconditioning technique is not so effective for large scale voxel finite element analysis because it requires more memory regions. Though, the use of converged displacement at the previous remodeling step was proved to be able to accelerate the convergence of conjugate gradient solver at the next step only when it is applied with the convergence criteria proposed by Rietbergen et al [7].

A complicated and detailed analysis models of bone microstructure with 4 million and 400 million d.o.f. (degrees of freedom) were analyzed with our approach until 50th remodeling step on PC-clusters. The 4 million d.o.f. voxel model of human proximal femur with 0.5mm resolution was constructed using serial images measured by medical X-ray CT and another 400 million d.o.f. model was made by subdividing the finite elements of femur with 0.5mm resolution into 5x5x5 voxel elements. The effectiveness of large scale finite element analysis was verified by comparing the remodeling results of these two models. Finally, shape adaptation analysis of human proximal femur with 400 million d.o.f. was analysed in 451 hours until 50th remodeling steps on 16 Pentium 4 processors.

Parallel large scale image based analysis make it possible to investigate the detailed remodeling features by using computer simulation in reasonable time. For accurate remodeling simulation, more than 50 steps are due to be performed in 400 million d.o.f. problem.

1

S.J. Hollister, N. Kikuchi, Homogenization Theory and Digital Imaging: A Basis for Study-ing the Mechanics and Design Principles of Bone Tissue, Biotechnol. & Bioeng., 43-7, 586-596. 1994. doi:10.1002/bit.260430708

2

B. van Rietbergen, H. Weinans, R. Huiskes, A. Odgaard, A New Method to Determine Trabecular Bone Elastic Properties and Loading Using Micromechanical Finite Element Models, J. Biomech., 28-1, 69-81, 1995. doi:10.1016/0021-9290(95)80008-5

3

T. Adachi, Y. Tomita, H. Sakaue, M. Tanaka, Simulation of Trabecular Surface Remodeling Based on Local Stress Nonuniformity, JSME Int. J., 40, 782-792, 1997.

4

T. Adachi, K. Tsubota, Y. Tomita, S.J. Hollister, Trabecular Surface Remodeling Simula-tion for Cancellous Bone Using Microstructural Voxel Finite Element Models, Trans. ASME, J. Biomech. Eng., 123-5, 403-409, 2001. doi:10.1115/1.1392315

5

T.J.R. Hughes, R.M. Ferenca, J.O. Hallquist, Large Scale Vectorized Implicit Calculations in Solid Mechanics on a Cray X-MP/48 Utilizing EBE Preconditioned Conjugate Gradients, Comp. Meth. Appl. Mech. Engrg., 61, 215-248, 1987. doi:10.1016/0045-7825(87)90005-3

6

G. Yagawa, Y. Nakabayashi and H. Okuda, Large-Scale Finite Element Fluid Analysis by Massively Parallel Processors, Special Issue of Parallel Computing on Parallel Computing in Applied Fluid Mechanics, 23, 1365-1377, 1997. doi:10.1016/S0167-8191(97)89284-8

7

van Rietbergen, B., Eckstein, F., Koller, B., Huiskes, R., Baaijens, F.P.T., Rüegsegger, P., Feasibility of FE analyses of human bones, Computer Methods in Biomechanics & Biomedical Engineering -3. New York: Gordon and Breach Science Publishers, 129-134, 2001.

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