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Computational simulation of internal bone remodeling

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Summary

A review of the state of the art in computational modeling and analysis of the mechanical behavior of living bone is given. Particular attention is placed on algorithms for the simulation of the stress or strain induced remodeling processes. A special remodeling algorithm is presented which allow the simulation of internal bone remodeling taking into account not only adaptation of the spatial distribution of the effective mass density, but also the adaptation of the orientation of the material axes and of the orientation dependent stiffness parameters. Such remodeling algorithms require a sound formulation of the constitutive relations of bony material. For this purpose some micro-macro mechanical descriptions of bone in its different microstructural configurations are discussed. In conjunction with the above mentioned remodeling algorithm a new unified material model is derived for describing the linear elastic, orthotropic behavior of bone in the full range of micro-structures of cancellous and cortical bone. The application of the novel remodeling algorithm is demonstrated in an example.

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References

  1. Ashman, R.B., Rho, J.Y. (1988), “Elastic Modulus of Trabecular Bone Material”,J. Biomech.,21, pp. 177–181.

    Article  Google Scholar 

  2. Beaupré, G.S. and Hayes W.C. (1985), “Finite Element Analysis of a Three-Dimensional Open-Cell Model for Trabecular Bone”,J. Biomed. Engng.,107, pp. 249–256.

    Google Scholar 

  3. Beaupré, G.S., Orr, T.E. and Carter, D.R. (1990), “An Approach for Time-Dependent Bone Modeling and Remodeling—Theoretical Development”,J. Orthop. Res.,8, pp. 651–661.

    Article  Google Scholar 

  4. Bucháček, K. (1990), “Nonequilibrium Bone Remodelling: Changes of Mass Density and of the Axes of Anisotropy”,Int. J. Engng. Sci.,28, pp. 1039–1044.

    Article  MATH  Google Scholar 

  5. Büchter, N. and Ramm, E. (1992), “Large Rotations in Structural Mechanics—Overview”, InNonlinear Analysis of Shells by Finite Elements, F.G. Rammerstorfer (Ed.), pp. 1–13, Springer-Verlag, Wien.

    Google Scholar 

  6. Carter, D.R. and Hayes, W.C. (1977), “The Compressive Behavior of Bone as a Two-Phase Porous Structure”,J. Bone Jt. Surg.,59-A, pp. 954–962.

    Google Scholar 

  7. Carter, D.R., Fyhrie, D.P. and Whalen, R.T. (1987), “Trabecular Bone Density and Loading History: Regulation of Connective Tissue Morphology by Mechanical Energy”,J. Biomech.,20, pp. 785–794.

    Article  Google Scholar 

  8. Carter, D.R., Orr, T.E. and Fyhrie, D.P. (1989), “Relationship between Loading History and Femoral Cancellous Bone Architecture”,J. Biomech.,22, pp. 231–244.

    Article  Google Scholar 

  9. Christensen, R.M. (1986), “Mechanics of Low Density Materials”,J. Mech. Phys. Solids,34, pp. 563–578.

    Article  Google Scholar 

  10. Cowin, S.C. and Hegedus, D.H. (1976), “Bone Remodeling I: Theory of Adaptive Elasticity”,J. Elasticity,6, pp. 313–326.

    MATH  MathSciNet  Google Scholar 

  11. Cowin, S.C. and nachlinger, R.R. (1978), “Bone Remodeling III: Uniqueness and Stability in Adaptive Elasticity Theory”,J. Elasticity,8, pp. 285–295.

    Article  MATH  MathSciNet  Google Scholar 

  12. Cowin, S.C. (1985), “The Relationship between the Elasticity Tensor and the Fabric Tensor”,Mechanics of Materials,4, pp. 137–147.

    Article  Google Scholar 

  13. Cowin, S.C. (1987), “Bone Remodeling of Diaphysial Surfaces by Torsional Loads: Theoretical Predictions”,J. Biomech.,20, pp. 1111–1120.

    Article  Google Scholar 

  14. Cowin, S.C., Sadegh, A.M. and Luo, G.M. (1992), “An Evolutionary Wolff's Law for Trabecular Architecture”,J. Biomech. Engng.,114, pp. 129–136.

    Google Scholar 

  15. Currey, J.D. (1964), “Three Analogies to Explain the Mechanical Properties of Bone”,Biorheology,2, pp. 1–10.

    Google Scholar 

  16. Firoozbakhsh, K. and Aleyaasin, M. (1989), “The Effect of Stress Concentration on Bone Remodeling: Theoretical Predictions”,J. Biomech. Engng.,111, pp. 355–360.

    Google Scholar 

  17. Firoozbakhsh, K., Aleyaasin, M. and Moneim, M.S. (1992), “Evolution of Bone Inhomogeneity Around a Hole in an Orthotropic Plate of Bone: Theoretical Predictions”,J. Biomech.,25, (4), pp. 387–394.

    Article  Google Scholar 

  18. Frost, H.M. (1964), “The Laws of Bone Structure”, Charles C. Thomas, Springfield, IL.

    Google Scholar 

  19. Fyhrie, D.P. and Carter, D.R. (1986), “A Unifying Principle Relating Stress to Trabecular Bone Morphology”,J. Orthop. Res.,4, pp. 304–317.

    Article  Google Scholar 

  20. Gibson, L.J. (1985), “The Mechanical Behavior of Cancellous Bone”,J. Biomech.,18, pp. 317–328.

    Article  Google Scholar 

  21. Gibson, L.J. and Ashby, M.F. (1988), “Cellular Solids: Structure and Properties”, Pergamon Press, Oxford.

    MATH  Google Scholar 

  22. Goldstein, S.A., Wilson, D.L., Sonstegard, D.A. and Matthews, L.S. (1983), “The Mechanical Properties of Human Tibial Trabecular Bone as a Function of Metaphyscal Location”,J. Biomech.,16, pp. 965–969.

    Article  Google Scholar 

  23. Goulet, R.W., Goldstein, S.A., Ciarelli, M.J., Kuhn, J.L., Brown, M.B., and Feldkamp, L.A. (1994), “The Relationship Between the Structural and Orthogonal Compressive Properties of Trabecular Bone”,J. Biomech.,27, pp. 375–389.

    Article  Google Scholar 

  24. Harrigan, T.P. and Hamilton, J.J. (1992a), “An Analytical and Numerical Study of the Stability of Bone Remodeling Theories: Dependence on Microstructural Stimulus”,J. Biomech.,25, pp. 477–488.

    Article  Google Scholar 

  25. Harrigan, T.P. and Hamilton, J.J. (1992b), “Optimality Conditions for Finite Element Simulation of Adaptive Bone Remodeling”,Int. J. Solids Struct.,29, pp. 2897–2906.

    Article  MATH  Google Scholar 

  26. Harrigan, T.P. and Hamilton, J.J. (1993), “Finite Element Simulation of Adaptive Bone Remodelling: A Stability Criterion and a Time Stepping Method”,Int. J. Num. Meth. Eng.,36, pp. 837–854.

    Article  Google Scholar 

  27. Hart, R.T. and Davy, D.T. (1989), “Thoories of Bone Modeling and Remodeling”, InBone Mechanics, S.C. Cowin (Ed.), CRC Press, Boca Raton, FL, pp. 253–277.

    Google Scholar 

  28. Hayes, W.C. and Snyder, B. (1989), “Toward a Quantitative Formulation of Wolff's Law of Trabecular Bone”, InMechanical Properties of Bone, S.C. Cowin (Ed.), AMD,45, ASME, New York, NY, pp. 43–68.

    Google Scholar 

  29. Hegedus, D.H. and Cowin, S.C. (1976), “Bone Remodeling II: Small Strain Adaptive Elasticity”,J. Elasticity,6, pp. 337–352.

    Article  MATH  MathSciNet  Google Scholar 

  30. Hogan, H.A. (1989), “Micromechanics Modeling of Haversian Cortical Bone Properties”,J. Biomech.,22, pp.549–556.

    Article  Google Scholar 

  31. Hollister, S.C., Kikuchi, N., Borodkin, J.L. and Goldstein, S.A. (1990), “A Study of Microstructural Analysis Methods: Implications for Modeling Trabecular Bone Microstructure”, InAdvances in Bioengineering, S.A. Goldstein (Ed.), BED. Vol.17, ASME, New York, NY, pp. 387–390.

    Google Scholar 

  32. Hollister, S.J., Fyhric, D.P., Jepsen, K.J. and Goldstein, S.A. (1991), “Application of Homogenization Theory to the Study of Trabecular Bone Mechanics”,J. Biomech.,24, pp. 825–839.

    Article  Google Scholar 

  33. Hollister, S.J., Brennan, J.M. and Kikuchi, N. (1994), “A Homogenization Sampling Procedure for relating Trabecular Bone Effective Stiffness and Tissue Level Stress”,J. Biomech.,27, pp. 433–444.

    Article  Google Scholar 

  34. Huiskes, R., Weinans, H., Grootenboer, H.J., Dalstra, M., Fudala, B. and Sloof, T.J. (1987), “Adaptive Bone-Remodeling Theory Applied to Prosthetic-Design Analysis”,J. Biomech.,22, pp. 1135–1150.

    Article  Google Scholar 

  35. Jacobs, C.R., Simo, J.C., Beaupré, G.S. and Carter, D.R. (1995), “Comparing an Optimal Global Efficiency Assumption to a Principal Stress-Based Formulation for the Simulation of Anisotropic Bone Adaptation to Mechanical Loading”, InProc. of the 1995 Bioengineering Conference, R.M. Hochmuthet al. (Eds.), BED, Vol.29, ASME, New York, NY, pp. 447–478.

    Google Scholar 

  36. Katz, J.L. (1971), “Hard Tissue as a Composite Material I. Bounds on the Elastic Behavior”,J. Biomech.,4, pp. 455–473.

    Article  Google Scholar 

  37. Katz, J.L. (1981), “Composite Material Models for Cortical Bone”, InMechanical Properties of Bone, S.C. Cowin (Ed.), AMD, Vol.45, ASME, New York, NY, pp. 171–184.

    Google Scholar 

  38. Katz, J.L. and Meunier, A. (1990), “A Generalized Method for Characterizing Elastic Anisotropy in Solid Living Tissues”,J. Mat. Sci. Mater. in Med.,1, pp. 1–8.

    Article  Google Scholar 

  39. Krach, W., Reiter, T.J., Rammerstorfer, F.G. and Zenz, P. (1995), “Computerunterstützte Vorhersage des Knochenumbaus am Beispiel von Knieendoprothesen”, InProc. d. 19. Jahrestagung der österr. Biornedizinischen Technik, Biomedizinische Technik Band 40, Ergänzungsband 2. Schiele & Schön, Berlin. GFR, pp. 78–80.

    Google Scholar 

  40. Kummer, B.K.F. (1972), “Biomechanics of Bone: Mechanical Properties, Functional Structures, Functional Adaptation”, InBiomechanics: Its Foundation and Objectives, Y.C. Fung (Ed.), Prentice-Hall, Englewood Cliffs, NJ, pp. 237–271.

    Google Scholar 

  41. Lees, S. and Davidson, C.L. (1977), “The Role of Collagen in the Elastic Properties of Calcified Tissues”,J. Biomech.,10, pp. 473–486.

    Article  Google Scholar 

  42. Lipson, S.F. and Katz, J.L. (1984), “The Relationship Between Elastic Properties and Microstructure of Bovine Cortical Bone”,J. Biomech.,17, pp. 231–240.

    Article  Google Scholar 

  43. Martin, R.B. and Burr, D.B. (1989), “Structure, Function and Adaptation of Compact Bone”, Raven Press, New York, NY.

    Google Scholar 

  44. Martin, R.B. (1984), “Porosity and Specific Surface of Bone”, in:CRC Critical Reviews in Biomedical Engng.,10, CRC Press, Boca Raton, FL, pp. 179–222.

    Google Scholar 

  45. McElhancy, J.H., Alem, N. and Roberts, V. (1970), “A Porous Block Model for Cancellous Bone”,ASME Publication 70-WA/BHF-2, pp. 1–9.

  46. Misra, J.C. and Samanta, S. (1987), “Effect of Material Damping on Bone Remodeling”,J. Biomech.,20(3), pp. 241–249.

    Article  Google Scholar 

  47. Müller, R. (1994), “3D Assessment and Analysis of Trabecular Bone Architecture”. PhD. Thesis, ETH Zürich, Zurich.

    Google Scholar 

  48. Pauwels, F. (1965), “Gesammelte Abhandlungen zur funktionellen Anatomie des Bewegungsapparates”, Springer-Verlag, Berlin.

    Google Scholar 

  49. Pedersen, P. (1989), “On Optimal Orientation of Orthotropic Materials”,Struct. Opt.,1, pp. 101–106.

    Article  Google Scholar 

  50. Pettermann, H. (1993), “Entwicklung eines Simulationsagorithmus der natürlichen Anpassung des orthotropen Knochenmaterials”, Diploma Thesis, Vienna University of Technology. Vienna.

    Google Scholar 

  51. Pettermann, H.E., Reiter, T.J. and Rammerstorfer, F.G. (1995), “Ein Simulationsalgorithmus der natürlichen Anpassung des orthotropen Knochenmaterials”, InProc. d. 19. Jahrestagung der österr. Biomedizinischen Technik, Biomedizinische Technik Band 40, Ergänzungsband 2, Schiele & Schön, Berlin, GFR, pp. 127–129.

    Google Scholar 

  52. Pickarski, K. (1973) “Analysis of Bone as a Composite Material”,Int. J. Engng. Sci.,11, pp. 557–565.

    Article  Google Scholar 

  53. Prendergast, P.J. and Taylor, D. (1992), “Design of Intramedullary Prostheses to Prevent Bone Loss: Predictions Based on Damage-Stimulated Remodelling”,J. Biomed. Engng.,14, pp. 499–506.

    Article  Google Scholar 

  54. Pugh, J.W., Rose, R.M. and Radin, E.L. (1973), “Elastic and Viscoclastic Properties of Trabecular Bone: Dependence on Structure”,J. Biomech.,6, pp. 475–485.

    Article  Google Scholar 

  55. Reilly, D.T., Burstein, A.H. and Frankel, V.H. (1974), “The Elastic Modulus for Bone”,J. Biomech.,7, pp. 271–275.

    Article  Google Scholar 

  56. Reiter, T.J., Rammerstorfer, F.G. and Böhm, H.J. (1990), “A Numerical Algorithm for the Simulation of Bone Remodelling”, InAdvances in Bioengineering, S.A. Goldstein (Ed.), BED,17, ASME, New York, NY, pp. 181–184.

    Google Scholar 

  57. Reiter, T.J. and Rammerstorfer, F.G. (1993), “Simulation of Natural Adaptation of Bone Material and Appli-cation in Optimum Composition Design”,Optimal Design with Advanced Materials, P. Pedersen (Ed.), Elsevier, Amsterdam, pp. 25–36.

    Google Scholar 

  58. Reiter, T.J., Rammerstorfer, F.G. and Böhm, H.J. (1993a), “Computation of Biomechanical Effect in Bone Caused by Implants”, InComputer Methods in Mechanics, Vol. II, W. Gilewski and L. Chodor (Eds.), Kielce University of Technology, Kielce, pp. 799–806.

    Google Scholar 

  59. Reiter, T.J., Rammerstorfer, F.G. and Böhm, H.J. (1993b), “Structural Design Improvement by Functional Adaptation”, InProc. of Structural Optimization 93, J. Herskovits (Ed.), Rio de Janciros, pp. 361–368.

  60. Reiter, T.J., Böhm, H.J., Krach, W., Pleschberger, M. and Rammerstorfer, F.G. (1994a), “Some Applications of the Finite Element Method in Biomechanical Stress Analysis”,Int. J. Comput. Appl. Technol.,7, pp. 233–241.

    Google Scholar 

  61. Reiter, T.J., Pettermann, H.E. and Rammerstorfer, F.G. (1994b), “Simulation of Bone Remodeling with Respect to Bone Anisotroy”, InProc. of the 8th International Conference on Biomedical Engineering, J.C.H. Goh and A. Nather (Eds), Singapore, pp. 201–203.

  62. Reiter, T.J. (1995), “Knochen— eine selbstoptimierende Struktur?” InEvolution und Optimierung: Strategien in Natur und Technik, U. Kull and E. Ramm and R. Reiner (Eds.), S. Hierzel Wissenschaftliche Verlagsgesellschaft Stuttgart, GFR, pp. 121–135.

    Google Scholar 

  63. Reiter, T.J. (1996), “Functional Adaptation of Bone and Application in Optimal Structural Design”, VDI-Berichte, Reihe 17, Nr. 145, VDI-Verlag, Düsseldorf, GFR.

    Google Scholar 

  64. Rho, J.Y., Ashman, R.B. and Turner, Ch.H. (1993), “Young's Modulus of Trabecular and Cortical Bone Material: Ultrasonic and Microtensile Measurements”,J. Biomech.,26, pp. 111–119.

    Article  Google Scholar 

  65. Rice, J.C., Cowin, S.C. and Bowman, J.A. (1988), “On the Dependence of the Elasticity and Strength of Cancellous Bone on Apparent Density”,J. Biomech.,21, pp. 155–168.

    Article  Google Scholar 

  66. Van Rietbergen, B., Weinans, H., Huiskes, R. and Odgaard, A. (1995), “A New Method to Determine Trabecular Bone Elastics Properties and Loading Using Micromechanical Finite Element Models”,J. Biomech.,28, pp. 69–81.

    Article  Google Scholar 

  67. Starke, G.R., Mercer, C.D., Spirakis, A., Martin, J.B. and Learmonth, I.D. (1992), “Some Aspects of Adaptive Bone Growth as a Result of Total Hip Joint Replacement”, InProc. 1992 ABAQUS User's Conference, HKS Inc., Providence, RI, pp. 483–496.

  68. Suquet, P.M. (1990), “Une méthode simplifiée pour le calcul des proprietés élastiques de matériaux hétérogènes à structure périodique”,C. R. Acad Sci. Paris, série II,311, pp. 769–774.

    MATH  MathSciNet  Google Scholar 

  69. Townsend, P.R., Raux, P., Rose, R.M., Miegel, R.E. and Radin, E.L. (1975). “The Distribution and Anisotropy of the Stiffnesses of Cancellous Bone in the Human Patella”,J. Biomech.,8, pp. 363–367.

    Article  Google Scholar 

  70. Turner, C.H., Cowin, S.C., Rho, J.Y., Ashman, R.B. and Rice, J.C. (1990), “The Fabric Dependence of the Orthotropic Elastic Constants of Cancellous Bone”,J. Biomech.,23, pp. 549–561.

    Article  Google Scholar 

  71. Van Buskirk, W.C. and Ashman, R.B. (1981), “The Elastic Moduli of Bone”, InThe Mechanical Properties of Bone, S.C. Cowin (Ed.), AMD,45, ASME, New York, pp. 131–143.

    Google Scholar 

  72. Viceconti, M. and Seireg, A. (1990), “A Generalized Procedure for Predicting Bone Mass Regulation by Mechanical Strain”,Calcif. Tiss. Int.,47, pp. 296–301.

    Article  Google Scholar 

  73. Whalen, R.T., Carter, D.R. and Steele, C.R. (1988), “Influence of Physical Activity on the Regulation of Bone Density”,J. Biomech.,21, pp. 825–837.

    Article  Google Scholar 

  74. Williams, J.L. and Lewis, J.L. (1982), “Properties of an Anisotropic Model of Cancellous Bone from the Proximal Tibial Epiphysis”,J. Biomech. Engng.,104, pp. 50–56.

    Google Scholar 

  75. Wolff, J. (1892),Das Gesetz der Transformation der Knochen. Hirschwald, Berlin.

    Google Scholar 

  76. Zysset, P.K. and Curnier, A. (1995), “An Alternative Model for Anisotropic Elasticity of Trabecular Bone”, InProc. of the 1995 Bioengineering Conference. R.M. Hochmuthet al. (Eds.), BED,29, ASME, New York, NY, pp. 359–360.

    Google Scholar 

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  1. T. J. Reiter

    Present address: VAI, Linz, Austria

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  1. Institute of Lightweight Structures and Aerospace Engineering, Vienna University of Technology, Austria, Gusshausstr. 27-29/E317, A-1040, Vienna, Austria

    H. E. Pettermann, T. J. Reiter & F. G. Rammerstorfer

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Pettermann, H.E., Reiter, T.J. & Rammerstorfer, F.G. Computational simulation of internal bone remodeling. ARCO 4, 295–323 (1997). https://doi.org/10.1007/BF02737117

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