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Numerical Modeling of Long Bone Adaptation due to Mechanical Loading: Correlation with Experiments

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Abstract

The process of external bone adaptation in cortical bone is modeled mathematically using finite element (FE) stress analysis coupled with an evolution model, in which adaptation response is triggered by mechanical stimulus represented by strain energy density. The model is applied to experiments in which a rat ulna is subjected to cyclic loading, and the results demonstrate the ability of the model to predict the bone adaptation response. The FE mesh is generated from micro-computed tomography (μCT) images of the rat ulna, and the stress analysis is carried out using boundary and loading conditions on the rat ulna obtained from the experiments [Robling, A. G., F. M. Hinant, D. B. Burr, and C. H. Turner. J. Bone Miner. Res. 17:1545–1554, 2002]. The external adaptation process is implemented in the model by moving the surface nodes of the FE mesh 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). 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 of Robling et al. (J. Bone Miner. Res. 17:1545–1554, 2002), 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.

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References

  1. ABAQUS Inc., ABAQUS Version 6.7 Documentation. 2008.

  2. Akhter, M. P., D. M. Raab, C. H. Turner, D. B. Kimmel, and R. R. Recker. Characterization of in vivo strain in the rat tibia during external application of a four-point bending load. J. Biomech. 25:1241–1246, 1992.

    Article  CAS  PubMed  Google Scholar 

  3. Altair Hypermesh. Altair HyperMesh Documentation. 2007.

  4. Amira. Amira User’s Guide. 2005.

  5. Beaupré, G. S., T. E. Orr, and D. R. Carter. An approach for time-dependent bone modeling and remodeling—theoretical development. J. Orthop. Res. 8:651–661, 1990.

    Article  PubMed  Google Scholar 

  6. Brebbia, C. A., and J. Dominguez. Boundary Elements: An Introductory Course. Southampton: WIT Press, 1989.

    Google Scholar 

  7. Burr, D. B., A. G. Robling, and C. H. Turner. Effects of biomechanical stress on bones in animals. Bone 30:781–786, 2002.

    Article  PubMed  Google Scholar 

  8. Carter, D. R., T. E. Orr, and D. P. Fyhrie. Relationships between loading history and femoral cancellous bone architecture. J. Biomech. 22:231–244, 1989.

    Article  CAS  PubMed  Google Scholar 

  9. Carter, D. R., M. C. H. Van der Meulen, and G. S. Beaupré. Mechanical factors in bone growth and development. Bone 18:S5–S10, 1996.

    Article  Google Scholar 

  10. Cowin, S. C. (ed.). Bone Mechanics Handbook. Boca Raton: CRC Press, 2001.

    Google Scholar 

  11. Cowin, S. C., and K. Firoozbakhsh. Bone remodeling of diaphysial surfaces under constant load: theoretical predictions. J. Biomech. 14:471–484, 1981.

    Article  CAS  PubMed  Google Scholar 

  12. Cowin, S. C., and D. H. Hegedus. Bone remodeling I: theory of adaptive elasticity. J. Elasticity 6:313–325, 1976.

    Article  Google Scholar 

  13. Discher, D. E., P. Janmey, and Y. Wang. Tissue cells feel and respond to the stiffness of their substrate. Science 310:1139–1143, 2005.

    Article  CAS  PubMed  Google Scholar 

  14. Doblaré, M., and J. M. García. Anisotropic bone remodelling model based on a continuum damage-repair theory. J. Biomech. 35:1–17, 2002.

    Article  PubMed  Google Scholar 

  15. Fridez, P., L. Rakotomanana, A. Terrier, and P. F. Leyvraz. Three dimensional model of external bone adaptation. Comput. Methods Biomech. Biomed. Eng. 2:189–196, 1998.

    Google Scholar 

  16. Frost, H. M. Skeletal structural adaptations to mechanical usage (SATMU): 1. Redefining Wolff’s law: the bone modeling problem. Anat. Rec. 226:403–413, 1990.

    Article  CAS  PubMed  Google Scholar 

  17. Frost, H. M. Skeletal structural adaptations to mechanical usage (SATMU): 2. Redefining Wolff’s law: the remodeling problem. Anat. Rec. 226:414–422, 1990.

    Article  CAS  PubMed  Google Scholar 

  18. Gross, T. S., S. Srinivasan, C. C. Liu, T. L. Clemens, and S. D. Bain. Noninvasive loading of the murine tibia: an in vivo model for the study of mechanotransduction. J. Bone Miner. Res. 17:493–501, 2002.

    Article  PubMed  Google Scholar 

  19. Han, Y., S. C. Cowin, B. M. Schaffler, and S. Weinbaum. Mechanotransduction and strain amplification in osteocyte cell processes. PNAS 101:16689–16694, 2004.

    Article  CAS  PubMed  Google Scholar 

  20. Hsieh, Y. F., and C. H. Turner. Effects of loading frequency on mechanically induced bone formation. J. Bone Miner. Res. 16:918–924, 2001.

    Article  CAS  PubMed  Google Scholar 

  21. Huiskes, R., R. Ruimerman, G. H. van Lenthe, and J. D. Janssen. Effects of mechanical forces on maintenance and adaptation of form in trabecular bone. Nature 405:704–706, 2000.

    Article  CAS  PubMed  Google Scholar 

  22. Huiskes, R., H. Weinans, H. J. Grootenboer, M. Dalstra, B. Fudala, and T. J. Slooff. Adaptive bone-remodeling theory applied to prosthetic-design analysis. J. Biomech. 20:1135–1150, 1987.

    Article  CAS  PubMed  Google Scholar 

  23. Knothe Tate, M. L. “Whither flows the fluid in bone?” An osteocyte’s perspective. J. Biomech. 36:1409–1424, 2003.

    Article  PubMed  Google Scholar 

  24. Knothe Tate, M. L., P. Niederer, and U. Knothe. In vivo tracer transport through the lacunocanalicular system of rat bone in an environment devoid of mechanical loading. Bone 22:107–117, 1998.

    Article  CAS  PubMed  Google Scholar 

  25. Lanyon, L. E., A. E. Goodship, C. J. Pye, and J. H. MacFie. Mechanically adaptive bone remodelling. J. Biomech. 15:141–154, 1982.

    Article  CAS  PubMed  Google Scholar 

  26. Lanyon, L. E., and C. T. Rubin. Static vs dynamic loads as an influence on bone remodelling. J. Biomech. 17:897–905, 1984.

    Article  CAS  PubMed  Google Scholar 

  27. Levenston, M. E., and D. R. Carter. An energy dissipation-based model for damage stimulated bone adaptation. J. Biomech. 31:579–586, 1998.

    Article  CAS  PubMed  Google Scholar 

  28. Martinez, G., J. M. Garcia Aznar, M. Doblare, and M. Cerrolaza. External bone remodeling through boundary elements and damage mechanics. Math. Comput. Simul. 73:183–199, 2006.

    Article  Google Scholar 

  29. McNamara, B. P., P. J. Prendergast, and D. Taylor. Prediction of bone adaptation in the ulnar-osteotomized sheep’s forelimb using an anatomical finite element model. J. Biomed. Eng. 14:209–216, 1992.

    Article  CAS  PubMed  Google Scholar 

  30. Pavalko, F. M., N. X. Chen, C. H. Turner, D. B. Burr, S. Atkinson, Y. Hsieh, J. Qiu, and R. L. Duncan. Fluid shear-induced mechanical signaling in MC3T3-E1 osteoblasts requires cytoskeleton-integrin interactions. Am. J. Physiol. Cell Physiol. 275:C1591–C1601, 1998.

    CAS  Google Scholar 

  31. Prendergast, P. J., and D. Taylor. Prediction of bone adaptation using damage accumulation. J. Biomech. 27:1067–1076, 1994.

    Article  CAS  PubMed  Google Scholar 

  32. Robling, A. G., D. B. Burr, and C. H. Turner. Partitioning a daily mechanical stimulus into discrete loading bouts improves the osteogenic response to loading. J. Bone Miner. Res. 15:1596–1602, 2000.

    Article  CAS  PubMed  Google Scholar 

  33. Robling, A. G., D. B. Burr, and C. H. Turner. Recovery periods restore mechanosensitivity to dynamically loaded bone. J. Exp. Biol. 204:3389–3399, 2001.

    CAS  PubMed  Google Scholar 

  34. Robling, A. G., F. M. Hinant, D. B. Burr, and C. H. Turner. Improved bone structure and strength after long-term mechanical loading is greatest if loading is separated into short bouts. J. Bone Miner. Res. 17:1545–1554, 2002.

    Article  PubMed  Google Scholar 

  35. Rubin, C., and L. Lanyon. Regulation of bone formation by applied dynamic loads. J. Bone Joint Surg. Am. 66:397–402, 1984.

    CAS  PubMed  Google Scholar 

  36. Rubin, C. T., and L. E. Lanyon. Regulation of bone mass by mechanical strain magnitude Calcif. Tissue Int. 37:411–417, 1985.

    Article  CAS  Google Scholar 

  37. Srinivasan, S., D. A. Weimer, S. C. Agans, S. D. Bain, and T. S. Gross. Low-magnitude mechanical loading becomes osteogenic when rest is inserted between each load cycle. J. Bone Miner. Res. 17:1613–1620, 2002.

    Article  PubMed  Google Scholar 

  38. Steck, R., P. Niederer, and M. L. Knothe Tate. A finite element analysis for the prediction of load-induced fluid flow and mechanochemical transduction in bone. J. Theor. Biol. 220:249–259, 2003.

    Article  CAS  PubMed  Google Scholar 

  39. Turner, C. H. Three rules for bone adaptation to mechanical stimuli. Bone 23:399–407, 1998.

    Article  CAS  PubMed  Google Scholar 

  40. Turner, C., M. Forwood, and M. Otter. Mechanotransduction in bone: do bone cells act as sensors of fluid flow? FASEB J. 8:875–878, 1994.

    CAS  PubMed  Google Scholar 

  41. van der Meulen, M. C. H., G. S. Beaupré, and D. R. Carter. Mechanobiologic influences in long bone cross-sectional growth. Bone 14:635–642, 1993.

    Article  PubMed  Google Scholar 

  42. Warden, S. J., and C. H. Turner. Mechanotransduction in cortical bone is most efficient at loading frequencies of 5–10 Hz. Bone 34:261–270, 2004.

    Article  CAS  PubMed  Google Scholar 

  43. Weinans, H., R. Huiskes, and H. J. Grootenboer. The behavior of adaptive bone-remodeling simulation models. J. Biomech. 25:1425–1441, 1992.

    Article  CAS  PubMed  Google Scholar 

  44. Weinbaum, S., S. C. Cowin, and Y. Zeng. A model for the excitation of osteocytes by mechanical loading-induced bone fluid shear stresses. J. Biomech. 27:339–360, 1994.

    Article  CAS  PubMed  Google Scholar 

  45. Zienkiewicz, O. C., and R. L. Taylor. The Finite Element Method. Volume 1—Basic Formulations and Linear Problems. London: McGraw-Hill, 1989.

    Google Scholar 

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Acknowledgments

We would like to thank Prof. Daniel Tortorelli of Department of Mechanical Science and Engineering, UIUC, for his help and advice on the smoothing filter approach, and Khanh Nguyen of Biomedical Engineering, Purdue, for the strain gage measurements. The support of the University of Illinois and the NIH through Grant AR046530 is also gratefully acknowledged.

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Authors and Affiliations

  1. Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USA

    Natarajan Chennimalai Kumar, Jonathan A. Dantzig & Iwona M. Jasiuk

  2. Department of Anatomy and Cell Biology, Indiana University School of Medicine, Indianapolis, IN, USA

    Alex G. Robling

  3. Department of Orthopedic Surgery and Biomedical Engineering, IUPUI, Indianapolis, IN, USA

    Charles H. Turner

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  1. Natarajan Chennimalai Kumar
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  2. Jonathan A. Dantzig
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Correspondence to Iwona M. Jasiuk.

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Chennimalai Kumar, N., Dantzig, J.A., Jasiuk, I.M. et al. Numerical Modeling of Long Bone Adaptation due to Mechanical Loading: Correlation with Experiments. Ann Biomed Eng 38, 594–604 (2010). https://doi.org/10.1007/s10439-009-9861-4

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