On a Constitutive Material Model to Capture Time Dependent Behavior of Cortical Bone
Affiliation(s)
1 Department of Prosthodontics, Faculty of Odontology, Malmo University, Malmo, Sweden. 2 DENTSPLY Implants, Molndal, Sweden. 3 Department of Applied Mechanics, Chalmers University of Technology, Goteborg, Sweden.
1 Department of Prosthodontics, Faculty of Odontology, Malmo University, Malmo, Sweden. 2 DENTSPLY Implants, Molndal, Sweden. 3 Department of Applied Mechanics, Chalmers University of Technology, Goteborg, Sweden.
ABSTRACT
It is commonly known that cortical bone exhibits viscoelastic-viscoplastic behavior which affects the biomechanical response when an implant is subjected to an external load. In addition, long term effects such as creep, relaxation and remodeling affect the success of the implant over time. Constitutive material models are commonly derived from data obtained in in vitro experiments. However during function, remodeling of bone greatly affects the bone material over time. Hence it is essential to include long term in vivo effects in a constitutive model of bone. This paper proposes a constitutive material model for cortical bone incorporating viscoelasticity, viscoplasticity, creep and remodeling to predict stress-strain at various strain rates as well as the behavior of bone over time in vivo. The rheological model and its parameters explain the behavior of bone subjected to longitudinal loading. By a proper set of model parameters, for a specific cortical bone, the present model can be used for prediction of the behavior of this bone under specific loading conditions. In addition simulation with the proposed model demonstrates excellent agreement to in vitro and in vivo experimental results in the literature.
It is commonly known that cortical bone exhibits viscoelastic-viscoplastic behavior which affects the biomechanical response when an implant is subjected to an external load. In addition, long term effects such as creep, relaxation and remodeling affect the success of the implant over time. Constitutive material models are commonly derived from data obtained in in vitro experiments. However during function, remodeling of bone greatly affects the bone material over time. Hence it is essential to include long term in vivo effects in a constitutive model of bone. This paper proposes a constitutive material model for cortical bone incorporating viscoelasticity, viscoplasticity, creep and remodeling to predict stress-strain at various strain rates as well as the behavior of bone over time in vivo. The rheological model and its parameters explain the behavior of bone subjected to longitudinal loading. By a proper set of model parameters, for a specific cortical bone, the present model can be used for prediction of the behavior of this bone under specific loading conditions. In addition simulation with the proposed model demonstrates excellent agreement to in vitro and in vivo experimental results in the literature.
Cite this paper
Halldin, A. , Ander, M. , Jacobsson, M. and Hansson, S. (2014) On a Constitutive Material Model to Capture Time Dependent Behavior of Cortical Bone. World Journal of Mechanics, 4, 348-361. doi: 10.4236/wjm.2014.411034.
Halldin, A. , Ander, M. , Jacobsson, M. and Hansson, S. (2014) On a Constitutive Material Model to Capture Time Dependent Behavior of Cortical Bone. World Journal of Mechanics, 4, 348-361. doi: 10.4236/wjm.2014.411034.
References
[1] McElhaney, J.H. (1966) Dynamic Response of Bone and Muscle Tissue. Journal of Applied Physiology, 21, 1231-1236. [2] Wood, J.L. (1971) Dynamic Response of Human Cranial Bone. Journal of Biomechanics, 4, 1-12.
http://dx.doi.org/10.1016/0021-9290(71)90010-8 [3] Iyo, T., Sasaki, N., Maki, Y. and Nakata, M. (2006) Mathematical Description of Stress Relaxation of Bovine Femoral Cortical Bone. Biorheology, 43, 117-132. [4] Crowninshield, R.D. and Pope, M.H. (1974) The Response of Compact Bone in Tension at Various Strain Rates. Annals of Biomedical Engineering, 2, 217-225.
http://dx.doi.org/10.1007/BF02368492 [5] Currey, J.D. (1975) The Effects of Strain Rate, Reconstruction and Mineral Content on Some Mechanical Properties of Bovine Bone. Journal of Biomechanics, 8, 81-86.
http://dx.doi.org/10.1016/0021-9290(75)90046-9 [6] Carter, D.R., Caler, W.E., Spengler, D.M. and Frankel, V.H. (1981) Fatigue Behavior of Adult Cortical Bone: The Influence of Mean Strain and Strain Range. Acta Orthopaedica Scandinavica, 52, 481-490.
http://dx.doi.org/10.3109/17453678108992136 [7] Sedlin, E.D. (1965) A Rheologic Model for Cortical Bone. A Study of the Physical Properties of Human Femoral Samples. Acta Orthopaedica Scandinavica, 36, 1-77.
http://dx.doi.org/10.3109/ort.1965.36.suppl-83.01 [8] Perren, S.M., Huggler, A., Russenberger, M., Allgower, M., Mathys, R., Schenk, R., Willenegger, H. and Muller, M.E. (1969) The Reaction of Cortical Bone to Compression. Acta Orthopaedica Scandinavica, S125, 19-29. [9] Reilly, D.T. and Burstein, A.H. (1974) Review Article. The Mechanical Properties of Cortical Bone. The Journal of bone and Joint Surgery, 56, 1001-1022. [10] Reilly, D.T. and Burstein, A.H. (1975) The Elastic and Ultimate Properties of Compact Bone Tissue. Journal of Biomechanics, 8, 393-405.
http://dx.doi.org/10.1016/0021-9290(75)90075-5 [11] Johnson, T.P., Socrate, S. and Boyce, M.C. (2010) A Viscoelastic, Viscoplastic Model of Cortical Bone Valid at Low and High Strain Rates. Acta Biomaterialia, 6, 4073-4080.
http://dx.doi.org/10.1016/j.actbio.2010.04.017 [12] Garcia, D., Zysset, P.K., Charlebois, M. and Curnier, A. (2009) A Three-Dimensional Elastic Plastic Damage Constitutive Law for Bone Tissue. Biomechanics and Modeling in Mechanobiology, 8, 149-165.
http://dx.doi.org/10.1007/s10237-008-0125-2 [13] Mercer, C., He, M.Y., Wang, R. and Evans, A.G. (2006) Mechanisms Governing the Inelastic Deformation of Cortical Bone and Application to Trabecular Bone. Acta Biomaterialia, 2, 59-68.
http://dx.doi.org/10.1016/j.actbio.2005.08.004 [14] Shunmugasamy, V.C., Gupta, N. and Coelho, P.G. (2010) High Strain Rate Response of Rabbit Femur Bones. Journal of Biomechanics, 43, 3044-3050.
http://dx.doi.org/10.1016/j.jbiomech.2010.06.034 [15] Rho, J.Y., Kuhn-Spearing, L. and Zioupos, P. (1998) Mechanical Properties and the Hierarchical Structure of Bone. Medical Engineering & Physics, 20, 92-102.
http://dx.doi.org/10.1016/S1350-4533(98)00007-1 [16] Hamed, E., Novitskaya, E., Li, J., Chen, P.Y., Jasiuk, I. and McKittrick, J. (2012) Elastic Moduli of Untreated, Demineralized and Deproteinized Cortical Bone: Validation of a Theoretical Model of Bone as an Interpenetrating Composite Material. Acta Biomaterialia, 8, 1080-1092.
http://dx.doi.org/10.1016/j.actbio.2011.11.010 [17] Currey, J.D. (2004) Tensile Yield in Compact Bone Is Determined by Strain, Post-Yield Behaviour by Mineral Content. Journal of Biomechanics, 37, 549-556.
http://dx.doi.org/10.1016/j.jbiomech.2003.08.008 [18] Melnis, A.E. and Knets, I.V. (1982) Effect of the Rate of Deformation on the Mechanical Properties of Compact Bone Tissue. Mechanics of Composite Materials, 18, 358-363.
http://dx.doi.org/10.1007/BF00604319 [19] Pithioux, M., Subit, D. and Chabrand, P. (2004) Comparison of Compact Bone Failure under Two Different Loading Rates: Experimental and Modelling Approaches. Medical Engineering and Physics, 26, 647-653.
http://dx.doi.org/10.1016/j.medengphy.2004.05.002 [20] Hansen, U., Zioupos, P., Simpson, R., Currey, J.D. and Hynd, D. (2008) The Effect of Strain Rate on the Mechanical Properties of Human Cortical Bone. Journal of Biomechanical Engineering, 130, 011011-011018.
http://dx.doi.org/10.1115/1.2838032 [21] Wright, T.M. and Hayes, W.C. (1976) Tensile Testing of Bone over a Wide Range of Strain Rates: Effects of Strain Rate, Microstructure and Density. Medical & Biological Engineering, 14, 671-680.
http://dx.doi.org/10.1007/BF02477046 [22] Currey, J.D. (1988) Strain Rate and Mineral Content in Fracture Models of Bone. Journal of Orthopaedic Research, 6, 32-38.
http://dx.doi.org/10.1002/jor.1100060105 [23] Currey, J.D. (1988) The Effect of Porosity and Mineral Content on the Young’s Modulus of Elasticity of Compact Bone. Journal of Biomechanics, 21, 131-139.
http://dx.doi.org/10.1016/0021-9290(88)90006-1 [24] McCalden, R.W., McGeough, J.A., Barker, M.B. and Court-Brown, C.M. (1993) Age-Related Changes in the Tensile Properties of Cortical Bone. The Relative Importance of Changes in Porosity, Mineralization, and Microstructure. The Journal of Bone and Joint Surgery, American Volume, 75, 1193-1205. [25] Zioupos, P., Hansen, U. and Currey, J.D. (2008) Microcracking Damage and the Fracture Process in Relation to Strain Rate in Human Cortical Bone Tensile Failure. Journal of Biomechanics, 41, 2932-2939.
http://dx.doi.org/10.1016/j.jbiomech.2008.07.025 [26] Iyo, T., Maki, Y., Sasaki, N. and Nakata, M. (2004) Anisotropic Viscoelastic Properties of Cortical Bone. Journal of Biomechanics, 37, 1433-1437.
http://dx.doi.org/10.1016/j.jbiomech.2003.12.023 [27] Melnis, A.E., Knets, I.V. and Moorlat, P.A. (1979) Deformation Behavior of Human Compact Bone Tissue upon Creep under Tensile Testing. Mechanics of Composite Materials, 15, 574-579.
http://dx.doi.org/10.1007/BF00605581 [28] Melnis, A.E. and Knets, I.V. (1982) Age-Related Changes in the Tensile Creep Properties of Human Compact Bone Tissue. Mechanics of Composite Materials, 17, 495-501.
http://dx.doi.org/10.1007/BF00605920 [29] Knet-s, I.V. and Vilks, Y.K. (1975) Creep of Compact Human Bony Tissue under Tension. Polymer Mechanics, 11, 543-547.
http://dx.doi.org/10.1007/BF00856779 [30] Fondrk, M., Bahniuk, E., Davy, D.T. and Michaels, C. (1988) Some Viscoplastic Characteristics of Bovine and Human Cortical Bone. Journal of Biomechanics, 21, 623-630.
http://dx.doi.org/10.1016/0021-9290(88)90200-X [31] Blumlein, H., Cordey, J., Schneider, U.A., Rahn, B.A. and Perren, S.M. (1977) Long-Term Measurements of Axial Force of Screws in Vivo for Osteosynthesis-Langzeitmessung der Axialkraft von Knochenschrauben in Vivo. Zeitschrift Fur Orthopadie Und Ihre Grenzgebiete, 115, 603-604. [32] Cordey, J., Blumlein, H., Ziegler, W. and Perren, S.M. (1976) Study of the Behavior in the Course of Time of the Holding Power of Cortical Screws in Vivo. Acta Orthopaedica Belgica, 42, 75-87. [33] Roberts, W.E., Turley, P.K., Brezniak, N. and Fielder, P.J. (1987) Implants: Bone Physiology and Metabolism. CDA Journal California Dental Association, 15, 54-61. [34] Perzyna, P. (1966) Fundamental Problems in Viscoplasticity. Advances in Applied Mechanics, 9, 243-377. [35] Perzyna, P. (1971) Thermodynamic Theory of Viscoplasticity. Advances in Applied Mechanics, 11, 313-354. [36] Ottosen, N.S. and Ristinmaa, M. (2005) The Mechanics of Constitutive Modeling. Elsevier, Amsterdam. [37] Norton, F.H. (1929) The Creep of Steel at High Temperatures. McGraw-Hill Book Company, New York. [38] Garcia, D. (2006) Elastic Plastic Damage Laws for Cortical Bone. Ph.D. Dissertation, Thèse No. 3435, Ecole Polytechnique Federale De Lausanne, Lusanne. [39] Melnis, A.E., Kregers, A.F. and Villerush, K.K. (1982) An Evaluation of Some Factors Affecting the Creep Properties of Human Compact Bone Tissue. Mechanics of Composite Materials, 17, 711-715.
http://dx.doi.org/10.1007/BF00605058 [40] Halldin, A., Jimbo, R., Johansson, C.B., Wennerberg, A., Jacobsson, M., Albrektsson, T. and Hansson, S. (2011) The Effect of Static Bone Strain on Implant Stability and Bone Remodeling. Bone, 49, 783-789.
http://dx.doi.org/10.1016/j.bone.2011.07.003 [41] Halldin, A., Jimbo, R., Johansson, C.B., Wennerberg, A., Jacobsson, M., Albrektsson, T. and Hansson, S. (2014) Implant Stability and Bone Remodeling after 3 and 13 Days of Implantation with an Initial Static Strain. Clinical Implant Dentistry and Related Research, 16, 383-393.
http://dx.doi.org/10.1111/cid.12000
[1] McElhaney, J.H. (1966) Dynamic Response of Bone and Muscle Tissue. Journal of Applied Physiology, 21, 1231-1236. [2] Wood, J.L. (1971) Dynamic Response of Human Cranial Bone. Journal of Biomechanics, 4, 1-12.
http://dx.doi.org/10.1016/0021-9290(71)90010-8 [3] Iyo, T., Sasaki, N., Maki, Y. and Nakata, M. (2006) Mathematical Description of Stress Relaxation of Bovine Femoral Cortical Bone. Biorheology, 43, 117-132. [4] Crowninshield, R.D. and Pope, M.H. (1974) The Response of Compact Bone in Tension at Various Strain Rates. Annals of Biomedical Engineering, 2, 217-225.
http://dx.doi.org/10.1007/BF02368492 [5] Currey, J.D. (1975) The Effects of Strain Rate, Reconstruction and Mineral Content on Some Mechanical Properties of Bovine Bone. Journal of Biomechanics, 8, 81-86.
http://dx.doi.org/10.1016/0021-9290(75)90046-9 [6] Carter, D.R., Caler, W.E., Spengler, D.M. and Frankel, V.H. (1981) Fatigue Behavior of Adult Cortical Bone: The Influence of Mean Strain and Strain Range. Acta Orthopaedica Scandinavica, 52, 481-490.
http://dx.doi.org/10.3109/17453678108992136 [7] Sedlin, E.D. (1965) A Rheologic Model for Cortical Bone. A Study of the Physical Properties of Human Femoral Samples. Acta Orthopaedica Scandinavica, 36, 1-77.
http://dx.doi.org/10.3109/ort.1965.36.suppl-83.01 [8] Perren, S.M., Huggler, A., Russenberger, M., Allgower, M., Mathys, R., Schenk, R., Willenegger, H. and Muller, M.E. (1969) The Reaction of Cortical Bone to Compression. Acta Orthopaedica Scandinavica, S125, 19-29. [9] Reilly, D.T. and Burstein, A.H. (1974) Review Article. The Mechanical Properties of Cortical Bone. The Journal of bone and Joint Surgery, 56, 1001-1022. [10] Reilly, D.T. and Burstein, A.H. (1975) The Elastic and Ultimate Properties of Compact Bone Tissue. Journal of Biomechanics, 8, 393-405.
http://dx.doi.org/10.1016/0021-9290(75)90075-5 [11] Johnson, T.P., Socrate, S. and Boyce, M.C. (2010) A Viscoelastic, Viscoplastic Model of Cortical Bone Valid at Low and High Strain Rates. Acta Biomaterialia, 6, 4073-4080.
http://dx.doi.org/10.1016/j.actbio.2010.04.017 [12] Garcia, D., Zysset, P.K., Charlebois, M. and Curnier, A. (2009) A Three-Dimensional Elastic Plastic Damage Constitutive Law for Bone Tissue. Biomechanics and Modeling in Mechanobiology, 8, 149-165.
http://dx.doi.org/10.1007/s10237-008-0125-2 [13] Mercer, C., He, M.Y., Wang, R. and Evans, A.G. (2006) Mechanisms Governing the Inelastic Deformation of Cortical Bone and Application to Trabecular Bone. Acta Biomaterialia, 2, 59-68.
http://dx.doi.org/10.1016/j.actbio.2005.08.004 [14] Shunmugasamy, V.C., Gupta, N. and Coelho, P.G. (2010) High Strain Rate Response of Rabbit Femur Bones. Journal of Biomechanics, 43, 3044-3050.
http://dx.doi.org/10.1016/j.jbiomech.2010.06.034 [15] Rho, J.Y., Kuhn-Spearing, L. and Zioupos, P. (1998) Mechanical Properties and the Hierarchical Structure of Bone. Medical Engineering & Physics, 20, 92-102.
http://dx.doi.org/10.1016/S1350-4533(98)00007-1 [16] Hamed, E., Novitskaya, E., Li, J., Chen, P.Y., Jasiuk, I. and McKittrick, J. (2012) Elastic Moduli of Untreated, Demineralized and Deproteinized Cortical Bone: Validation of a Theoretical Model of Bone as an Interpenetrating Composite Material. Acta Biomaterialia, 8, 1080-1092.
http://dx.doi.org/10.1016/j.actbio.2011.11.010 [17] Currey, J.D. (2004) Tensile Yield in Compact Bone Is Determined by Strain, Post-Yield Behaviour by Mineral Content. Journal of Biomechanics, 37, 549-556.
http://dx.doi.org/10.1016/j.jbiomech.2003.08.008 [18] Melnis, A.E. and Knets, I.V. (1982) Effect of the Rate of Deformation on the Mechanical Properties of Compact Bone Tissue. Mechanics of Composite Materials, 18, 358-363.
http://dx.doi.org/10.1007/BF00604319 [19] Pithioux, M., Subit, D. and Chabrand, P. (2004) Comparison of Compact Bone Failure under Two Different Loading Rates: Experimental and Modelling Approaches. Medical Engineering and Physics, 26, 647-653.
http://dx.doi.org/10.1016/j.medengphy.2004.05.002 [20] Hansen, U., Zioupos, P., Simpson, R., Currey, J.D. and Hynd, D. (2008) The Effect of Strain Rate on the Mechanical Properties of Human Cortical Bone. Journal of Biomechanical Engineering, 130, 011011-011018.
http://dx.doi.org/10.1115/1.2838032 [21] Wright, T.M. and Hayes, W.C. (1976) Tensile Testing of Bone over a Wide Range of Strain Rates: Effects of Strain Rate, Microstructure and Density. Medical & Biological Engineering, 14, 671-680.
http://dx.doi.org/10.1007/BF02477046 [22] Currey, J.D. (1988) Strain Rate and Mineral Content in Fracture Models of Bone. Journal of Orthopaedic Research, 6, 32-38.
http://dx.doi.org/10.1002/jor.1100060105 [23] Currey, J.D. (1988) The Effect of Porosity and Mineral Content on the Young’s Modulus of Elasticity of Compact Bone. Journal of Biomechanics, 21, 131-139.
http://dx.doi.org/10.1016/0021-9290(88)90006-1 [24] McCalden, R.W., McGeough, J.A., Barker, M.B. and Court-Brown, C.M. (1993) Age-Related Changes in the Tensile Properties of Cortical Bone. The Relative Importance of Changes in Porosity, Mineralization, and Microstructure. The Journal of Bone and Joint Surgery, American Volume, 75, 1193-1205. [25] Zioupos, P., Hansen, U. and Currey, J.D. (2008) Microcracking Damage and the Fracture Process in Relation to Strain Rate in Human Cortical Bone Tensile Failure. Journal of Biomechanics, 41, 2932-2939.
http://dx.doi.org/10.1016/j.jbiomech.2008.07.025 [26] Iyo, T., Maki, Y., Sasaki, N. and Nakata, M. (2004) Anisotropic Viscoelastic Properties of Cortical Bone. Journal of Biomechanics, 37, 1433-1437.
http://dx.doi.org/10.1016/j.jbiomech.2003.12.023 [27] Melnis, A.E., Knets, I.V. and Moorlat, P.A. (1979) Deformation Behavior of Human Compact Bone Tissue upon Creep under Tensile Testing. Mechanics of Composite Materials, 15, 574-579.
http://dx.doi.org/10.1007/BF00605581 [28] Melnis, A.E. and Knets, I.V. (1982) Age-Related Changes in the Tensile Creep Properties of Human Compact Bone Tissue. Mechanics of Composite Materials, 17, 495-501.
http://dx.doi.org/10.1007/BF00605920 [29] Knet-s, I.V. and Vilks, Y.K. (1975) Creep of Compact Human Bony Tissue under Tension. Polymer Mechanics, 11, 543-547.
http://dx.doi.org/10.1007/BF00856779 [30] Fondrk, M., Bahniuk, E., Davy, D.T. and Michaels, C. (1988) Some Viscoplastic Characteristics of Bovine and Human Cortical Bone. Journal of Biomechanics, 21, 623-630.
http://dx.doi.org/10.1016/0021-9290(88)90200-X [31] Blumlein, H., Cordey, J., Schneider, U.A., Rahn, B.A. and Perren, S.M. (1977) Long-Term Measurements of Axial Force of Screws in Vivo for Osteosynthesis-Langzeitmessung der Axialkraft von Knochenschrauben in Vivo. Zeitschrift Fur Orthopadie Und Ihre Grenzgebiete, 115, 603-604. [32] Cordey, J., Blumlein, H., Ziegler, W. and Perren, S.M. (1976) Study of the Behavior in the Course of Time of the Holding Power of Cortical Screws in Vivo. Acta Orthopaedica Belgica, 42, 75-87. [33] Roberts, W.E., Turley, P.K., Brezniak, N. and Fielder, P.J. (1987) Implants: Bone Physiology and Metabolism. CDA Journal California Dental Association, 15, 54-61. [34] Perzyna, P. (1966) Fundamental Problems in Viscoplasticity. Advances in Applied Mechanics, 9, 243-377. [35] Perzyna, P. (1971) Thermodynamic Theory of Viscoplasticity. Advances in Applied Mechanics, 11, 313-354. [36] Ottosen, N.S. and Ristinmaa, M. (2005) The Mechanics of Constitutive Modeling. Elsevier, Amsterdam. [37] Norton, F.H. (1929) The Creep of Steel at High Temperatures. McGraw-Hill Book Company, New York. [38] Garcia, D. (2006) Elastic Plastic Damage Laws for Cortical Bone. Ph.D. Dissertation, Thèse No. 3435, Ecole Polytechnique Federale De Lausanne, Lusanne. [39] Melnis, A.E., Kregers, A.F. and Villerush, K.K. (1982) An Evaluation of Some Factors Affecting the Creep Properties of Human Compact Bone Tissue. Mechanics of Composite Materials, 17, 711-715.
http://dx.doi.org/10.1007/BF00605058 [40] Halldin, A., Jimbo, R., Johansson, C.B., Wennerberg, A., Jacobsson, M., Albrektsson, T. and Hansson, S. (2011) The Effect of Static Bone Strain on Implant Stability and Bone Remodeling. Bone, 49, 783-789.
http://dx.doi.org/10.1016/j.bone.2011.07.003 [41] Halldin, A., Jimbo, R., Johansson, C.B., Wennerberg, A., Jacobsson, M., Albrektsson, T. and Hansson, S. (2014) Implant Stability and Bone Remodeling after 3 and 13 Days of Implantation with an Initial Static Strain. Clinical Implant Dentistry and Related Research, 16, 383-393.
http://dx.doi.org/10.1111/cid.12000