Open forward and inverse problems in theoretical modeling of bone tissue adaptation

Journal of the Mechanical Behavior of Biomedical Materials

Volumn 27, Issue , 2013, Pages 249-261

  Zadpoor, Amir Abbas ^a  

^a DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Adaptation models; Bone remodeling; Finite element; Orthopedics; Review

Indexed keywords

ADAPTATION MODELS; BONE REMODELING; BONE TISSUE; FORWARD PROBLEM; THEORETICAL MODELING;

DIFFERENTIAL EQUATIONS; FINITE ELEMENT METHOD; INVERSE PROBLEMS; ORTHOPEDICS; REVIEWS; TISSUE;

BONE;

ADAPTATION; ANATOMIC MODEL; ARTICLE; BIOMECHANICS; BONE DENSITY; BONE REGENERATION; BONE REMODELING; BONE TISSUE; BONE TISSUE ADAPTATION ADAPTATION; CALCIUM HOMEOSTASIS; COMPUTER ASSISTED TOMOGRAPHY; DUAL ENERGY X RAY ABSORPTIOMETRY; GROUND REACTION FORCE; HOMEOSTASIS; HUMAN; IMAGE PROCESSING; MATHEMATICAL MODEL; MUSCULOSKELETAL FUNCTION; NUCLEAR MAGNETIC RESONANCE IMAGING; ORTHOPEDICS; OSTEOBLAST; OSTEOCLAST ACTIVITY; PHYSICAL ACTIVITY; PRIORITY JOURNAL; RUNNING; SIMULATION; THEORETICAL MODEL; TISSUE REACTION; WALKING;

ADAPTATION MODELS; BONE REMODELING; FINITE ELEMENT; ORTHOPEDICS; REVIEW;

ADAPTATION, PHYSIOLOGICAL; ANIMALS; BONE AND BONES; BONE DENSITY; HUMANS; MODELS, BIOLOGICAL; WEIGHT-BEARING;

EID: 84884289231     PISSN: 17516161     EISSN: 18780180     Source Type: Journal    
DOI: 10.1016/j.jmbbm.2013.05.017     Document Type: Article

References (122)

1. Audu M., Davy D. The influence of muscle model complexity in musculoskeletal motion modeling. Journal of Biomechanical Engineering 1985, 107:147.
2. Bagge M. A model of bone adaptation as an optimization process. Journal of Biomechanics 2000, 33:1349-1357.
3. Baiotto S., Labat B., Vico L., Zidi M. Bone remodeling regulation under unloading conditions: Numerical investigations. Computers in Biology and Medicine 2009, 39:46-52.
4. Bergmann G., Deuretzbacher G., Heller M., Graichen F., Rohlmann A., Strauss J., Duda G. Hip contact forces and gait patterns from routine activities. Journal of Biomechanics 2001, 34:859-871.
5. Bona M.A., Martin L.D., Fischer K.J. A contact algorithm for density-based load estimation. Journal of Biomechanics 2006, 39:636-644.
6. Bona M.A., Martin L.D., Fischer K.J. Density-based load estimation using two-dimensional finite element models: a parametric study. Computer Methods in Biomechanics and Biomedical Engineering 2006, 9:221-229.
7. Campoli G., Weinans H., Zadpoor A.A. Computational load estimation of the femur. Journal of the Mechanical Behavior of Biomedical Materials 2012, 10:108-119.
8. Carriero A., Jonkers I., Shefelbine S.J. Mechanobiological prediction of proximal femoral deformities in children with cerebral palsy. Computer Methods in Biomechanics and Biomedical Engineering 2011, 14:253-262.
9. Carter D., Fyhrie D., Whalen R. Trabecular bone density and loading history: regulation of connective tissue biology by mechanical energy. Journal of Biomechanics 1987, 20:789-794.
10. Carter D., Orr T., Fyhrie D. Relationships between loading history and femoral cancellous bone architecture. Journal of Biomechanics 1989, 22:231-244.
11. Charlton I.W., Johnson G.R. Application of spherical and cylindrical wrapping algorithms in a musculoskeletal model of the upper limb. Journal of biomechanics 2001, 34:1209-1216.
12. Chen G., Pettet G.J., Pearcy M., McElwain D.L.S. Modelling external bone adaptation using evolutionary structural optimisation. Biomechanics and Modeling in Mechanobiology 2007, 6:275-285.
13. Christen P., van Rietbergen B., Lambers F.M., Müller R., Ito K. Bone morphology allows estimation of loading history in a murine model of bone adaptation. Biomechanics and Modeling in Mechanobiology 2012, 11:483-492.
14. Cowin S., Hegedus D. Bone remodeling I: theory of adaptive elasticity. Journal of Elasticity 1976, 6:313-326.
15. Cowin S., Nachlinger R.R. Bone remodeling III: uniqueness and stability in adaptive elasticity theory. Journal of Elasticity 1978, 8:285-295.
16. Cox L., Van Rietbergen B., van Donkelaar C., Ito K. Analysis of bone architecture sensitivity for changes in mechanical loading, cellular activity, mechanotransduction, and tissue properties. Biomechanics and Modeling in Mechanobiology 2011, 10:701-712.
17. Doblare M. Anisotropic bone remodelling model based on a continuum damage-repair theory. Journal of Biomechanics 2002, 35:1-17.
18. Doblare M., Garcia J. Application of an anisotropic bone-remodelling model based on a damage-repair theory to the analysis of the proximal femur before and after total hip replacement. Journal of Biomechanics 2001, 34:1157-1170.
19. Dul J., Johnson G., Shiavi R., Townsend M. Muscular synergism- II. A minimum-fatigue criterion for load sharing between synergistic muscles. Journal of Biomechanics 1984, 17:675-684.
20. Dul J., Townsend M., Shiavi R., Johnson G. Muscular synergism- I. On criteria for load sharing between synergistic muscles. Journal of Biomechanics 1984, 17:663-673.
21. Fernandes P., Rodrigues H., Jacobs C. A model of bone adaptation using a global optimisation criterion based on the trajectorial theory of Wolff. Computer Methods in Biomechanics and Biomedical Engineering 1999, 2:125-138.
22. Fischer K.J., Bastidas J.A., Pfaeffle H.J., Towers J.D. A method for estimating relative bone loads from CT data with application to the radius and the ulna. CMES-Computer Modeling in Engineering and Sciences 2003, 4:397-403.
23. Fischer K.J., Eckstein F., Decker C. Density-based load estimation predicts altered femoral load directions for coxa vara and coxa valga. Journal of Musculoskeletal Research 1999, 3:83-92.
24. Fischer K.J., Jacobs C.R., Carter D.R. Computational method for determination of bone and joint loads using bone-density distributions. Journal of Biomechanics 1995, 28:1127-1135.
25. Fischer K.J., Jacobs C.R., Levenston M.E., Cody D.D., Carter D.R. Bone load estimation for the proximal femur using single energy quantitative CT data. Computer Methods in Biomechanics and Biomedical Engineering 1998, 1:233-245.
26. Garcia-Aznar J., Rueberg T., Doblare M. A bone remodelling model coupling microdamage growth and repair by 3D BMU-activity. Biomechanics and Modeling in Mechanobiology 2005, 4:147-167.
27. Graichen F., Bergmann G., Rohlmann A. Hip endoprosthesis for in vivo measurement of joint force and temperature. Journal of Biomechanics 1999, 32:1113-1117.
28. Guldberg R.E., Richards M., Caldwell N.J., Kuelske C.L., Goldstein S.A. Trabecular bone adaptation to variations in porous-coated implant topology. Journal of Biomechanics 1997, 30:147-153.
29. Hambli R., Rieger R. Physiologically based mathematical model of transduction of mechanobiological signals by osteocytes. Biomechanics and Modeling in Mechanobiology 2012, 11:83-93.
30. Harrigan T., Hamilton J. Optimality conditions for finite element simulation of adaptive bone remodeling. International Journal of Solids and Structures 1992, 29:2897-2906.
31. Harrigan T.P., Hamilton J.J. An analytical and numerical study of the stability of bone remodelling theories: dependence on microstructural stimulus. Journal of Biomechanics 1992, 25:477-488.
32. Harrigan T.P., Hamilton J.J. Bone remodeling and structural optimization. Journal of Biomechanics 1994, 27:323-328.
33. Harrigan T.P., Hamilton J.J. Necessary and sufficient conditions for global stability and uniqueness in finite element simulations of adaptive bone remodeling. International Journal of Solids and Structures 1994, 31:97-107.
34. Hazelwood S.J., Martin R.B., Rashid M.M., Rodrigo J.J. A mechanistic model for internal bone remodeling exhibits different dynamic responses in disuse and overload. Journal of Biomechanics 2001, 34:299.
35. Heimann T., Meinzer H.P. Statistical shape models for 3D medical image segmentation: a review. Medical Image Analysis 2009, 13:543.
36. Hernandez C.J., Beaupre G.S., Carter D.R. A model of mechanobiologic and metabolic influences on bone adaptation. Development 2000, 37:235-244.
37. Herrera A., Panisello J., Ibarz E., Cegoñino J., Puértolas J., Gracia L. Comparison between DEXA and finite element studies in the long-term bone remodeling of an anatomical femoral stem. Journal of Biomechanical Engineering 2009, 131:041013.
38. Hof A., Van Den Berg J. EMG to force processing I: An electrical analogue of the Hill muscle model. Journal of Biomechanics 1981, 14:747-758.
39. Hof A., Van den Berg J. EMG to force processing II: Estimation of parameters of the Hill muscle model for the human triceps surae by means of a calfergometer. Journal of Biomechanics 1981, 14:759-770.
40. Hollister S.J., Kikuchi N., Goldstein S.A. Do bone ingrowth processes produce a globally optimized structure?. Journal of biomechanics 1993, 26:391-407.
41. Hong J., Hipp J., Mulkern R., Jaramillo D., Snyder B. Magnetic resonance imaging measurements of bone density and cross-sectional geometry. Calcified Tissue International 2000, 66:74-78.
42. Hoy M.G., Zajac F.E., Gordon M.E. A musculoskeletal model of the human lower extremity: the effect of muscle, tendon, and moment arm on the moment-angle relationship of musculotendon actuators at the hip, knee, and ankle. Journal of biomechanics 1990, 23:157-169.
43. Huijing P.A., Yaman A., Ozturk C., Yucesoy C.A. Effects of knee joint angle on global and local strains within human triceps surae muscle: MRI analysis indicating in vivo myofascial force transmission between synergistic muscles. Surgical and Radiologic Anatomy 2011, 33:869-879.
44. Huiskes R. Validation of adaptive bone-remodeling simulation models. Studies in Health Technology and Informatics 1997, 33-50.
45. Huiskes R., Ruimerman R., van Lenthe G.H., Janssen J.D. Effects of mechanical forces on maintenance and adaptation of form in trabecular bone. Nature 2000, 405:704-706.
46. Huiskes R., Weinans H., Dalstra M. Adaptive bone remodeling and biomechanical design considerations. Orthopedics 1989, 12:1255-1267.
47. Huiskes R., Weinans H., Grootenboer H., Dalstra M., Fudala B., Slooff T. Adaptive bone-remodeling theory applied to prosthetic-design analysis. Journal of Biomechanics 1987, 20:1135-1150.
48. Huiskes R., Weinans H., Van Rietbergen B. The relationship between stress shielding and bone resorption around total hip stems and the effects of flexible materials. Clinical Orthopaedics and Related Research 1992, 274:124.
49. Jang I.G., Kim I.Y. Computational study of Wolff's law with trabecular architecture in the human proximal femur using topology optimization. Journal of Biomechanics 2008, 41:2353-2361.
50. Jang I.G., Kim I.Y. Application of design space optimization to bone remodeling simulation of trabecular architecture in human proximal femur for higher computational efficiency. Finite Elements in Analysis and Design 2010, 46:311-319.
51. Jang I.G., Kim I.Y., Kwak B.M. Analogy of strain energy density based bone-remodeling algorithm and structural topology optimization. Journal of Biomechanical Engineering 2009, 131:011012.
52. Jinha A., Ait-Haddou R., Herzog W. Predictions of co-contraction depend critically on degrees-of-freedom in the musculoskeletal model. Journal of Biomechanics 2006, 39:1145.
53. Kabel J., Van Rietbergen B., Odgaard A., Huiskes R. Constitutive relationships of fabric, density, and elastic properties in cancellous bone architecture. Bone 1999, 25:481-486.
54. Kim H.J., Fernandez J.W., Akbarshahi M., Walter J.P., Fregly B.J., Pandy M.G. Evaluation of predicted knee-joint muscle forces during gait using an instrumented knee implant. Journal of Orthopaedic Research 2009, 27:1326-1331.
55. Kroger H., Vainio P., Nieminen J., Kotaniemi A. Comparison of different models for interpreting bone mineral density measurements using DXA and MRI technology. Bone 1995, 17:157-159.
56. Lemaire V., Tobin F.L., Greller L.D., Cho C.R., Suva L.J. Modeling the interactions between osteoblast and osteoclast activities in bone remodeling. Journal of Theoretical Biology 2004, 229:293-309.
57. Lloyd D., Buchanan T. A model of load sharing between muscles and soft tissues at the human knee during static tasks. Journal of Biomechanical Engineering 1996, 118:367.
58. Lloyd D.G., Besier T.F. An EMG-driven musculoskeletal model to estimate muscle forces and knee joint moments in vivo. Journal of Biomechanics 2003, 36:765-776.
59. Martínez-Reina J., García-Aznar J., Dominguez J., Doblaré M. A bone remodelling model including the directional activity of BMUs. Biomechanics and Modeling in Mechanobiology 2009, 8:111-127.
60. McNamara L.M., Prendergast P.J. Bone remodelling algorithms incorporating both strain and microdamage stimuli. Journal of Biomechanics 2007, 40:1381-1391.
61. Meijer K., Grootenboer H., Koopman H., van Der Linden B., Huijing P.A. A Hill type model of rat medial gastrocnemius muscle that accounts for shortening history effects. Journal of Biomechanics 1998, 31:555-563.
62. Morgan E.F., Bayraktar H.H., Keaveny T.M. Trabecular bone modulus-density relationships depend on anatomic site. Journal of Biomechanics 2003, 36:897-904.
63. Mullender M., Huiskes R. Proposal for the regulatory mechanism of Wolff's law. Journal of Orthopaedic Research 1995, 13:503-512.
64. Mullender M., Huiskes R., Weinans H. A physiological approach to the simulation of bone remodeling as a self-organizational control process. Journal of Biomechanics 1994, 27:1389-1394.
65. Mullender M., Van Rietbergen B., Rüegsegger P., Huiskes R. Effect of mechanical set point of bone cells on mechanical control of trabecular bone architecture. Bone 1998, 22:125-131.
66. Newitt D., Majumdar S., Van Rietbergen B., Von Ingersleben G., Harris S., Genant H., Chesnut C., Garnero P., MacDonald B. In vivo assessment of architecture and micro-finite element analysis derived indices of mechanical properties of trabecular bone in the radius. Osteoporosis International 2002, 13:6-17.
67. Nikooyan A.A., Zadpoor A.A. An improved cost function for modeling of muscle activity during running. Journal of Biomechanics 2011, 44:984-987.
68. Nikooyan A.A., Zadpoor A.A. Mass-spring-damper modelling of the human body to study running and hopping- an overview. Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine 2011, 225:1121-1135.
69. Nikooyan A.A., Zadpoor A.A. Effects of muscle fatigue on the ground reaction force and soft-tissue vibrations during running: a model study. IEEE Transactions on Biomedical Engineering 2012, 59:797-804.
70. Odgaard A., Kabel J., van Rietbergen B., Dalstra M., Huiskes R. Fabric and elastic principal directions of cancellous bone are closely related. Journal of Biomechanics 1997, 30:487-496.
71. Pandy M.G., Sasaki K., Kim S. A three-dimensional musculoskeletal model of the human knee joint. Part 1: theoretical construct. Computer Methods in Biomechanics and Biomedical Engineering 1998, 1:87.
72. Payten W., Law M. Generalized shape optimization using stress constraints under multiple load cases. Structural and Multidisciplinary Optimization 1998, 15:269-274.
73. Peter B., Ramaniraka N., Rakotomanana L., Zambelli P., Pioletti D.P. Peri-implant bone remodeling after total hip replacement combined with systemic alendronate treatment: a finite element analysis. Computer Methods in Biomechanics and Biomedical Engineering 2004, 7:73-78.
74. Peterson M.C., Riggs M.M. A physiologically based mathematical model of integrated calcium homeostasis and bone remodeling. Bone 2010, 46:49-63.
75. Pettermann H., Reiter T.J., Rammerstorfer F.G. Computational simulation of internal bone remodeling. Archives of Computational Methods in Engineering 1997, 4:295-323.
76. Pioletti D.P., Rakotomanana L.R. Can the increase of bone mineral density following bisphosphonates treatments be explained by biomechanical considerations?. Clinical Biomechanics 2004, 19:170-174.
77. Pivonka P., Zimak J., Smith D.W., Gardiner B.S., Dunstan C.R., Sims N.A., John Martin T., Mundy G.R. Model structure and control of bone remodeling: A theoretical study. Bone 2008, 43:249-263.
78. Praagman M., Chadwick E., Van Der Helm F., Veeger H. The relationship between two different mechanical cost functions and muscle oxygen consumption. Journal of Biomechanics 2006, 39:758-765.
79. Prendergast P. Finite element models in tissue mechanics and orthopaedic implant design. Clinical Biomechanics 1997, 12:343-366.
80. Prendergast P., Taylor D. Prediction of bone adaptation using damage accumulation. Journal of Biomechanics 1994, 27:1067-1076.
81. Ramtani S., Garcia J., Doblare M. Computer simulation of an adaptive damage-bone remodeling law applied to three unit-bone bars structure. Computers in Biology and Medicine 2004, 34:259-273.
82. Ramtani S., Zidi M. A theoretical model of the effect of continuum damage on a bone adaptation model. Journal of Biomechanics 2001, 34:471-479.
83. Raposo J., Sobrinho L., Ferreira H. A minimal mathematical model of calcium homeostasis. Journal of Clinical Endocrinology & Metabolism 2002, 87:4330-4340.
84. Rho J.-Y., Kuhn-Spearing L., Zioupos P. Mechanical properties and the hierarchical structure of bone. Medical Engineering & Physics 1998, 20:92-102.
85. Rho J.Y., Roy M.E., Tsui T.Y., Pharr G.M. Elastic properties of microstructural components of human bone tissue as measured by nanoindentation. Journal of Biomedical Materials Research 1999, 45:48-54.
86. Rieger R., Hambli R., Jennane R. Modeling of biological doses and mechanical effects on bone transduction. Journal of Theoretical Biology 2011, 274:36-42.
87. Ruimerman R., Hilbers P., Van Rietbergen B., Huiskes R. A theoretical framework for strain-related trabecular bone maintenance and adaptation. Journal of Biomechanics 2005, 38:931-941.
88. Ruimerman R., Huiskes R. Development of a unifying theory for mechanical adaptation and maintenance of trabecular bone. Theoretical Issues in Ergonomics Science 2005, 6:225-238.
89. Ruimerman R., van Rietbergen B., Hilbers P., Huiskes R. A 3-dimensional computer model to simulate trabecular bone metabolism. Biorheology 2003, 40:315-320.
90. Shefelbine S.J., Augat P., Claes L., Simon U. Trabecular bone fracture healing simulation with finite element analysis and fuzzy logic. Journal of Biomechanics 2005, 38:2440-2450.
91. Shefelbine S.J., Carter D.R. Mechanobiological predictions of femoral anteversion in cerebral palsy. Annals of Biomedical Engineering 2004, 32:297-305.
92. Skedros J.G., Dayton M.R., Sybrowsky C.L., Bloebaum R.D., Bachus K.N. Are uniform regional safety factors an objective of adaptive modeling/remodeling in cortical bone?. Journal of Experimental Biology 2003, 206:2431.
93. Skinner H.B., Kilgus D.J., Keyak J., Shimaoka E.E., Kim A.S., Tipton J.S. Correlation of computed finite element stresses to bone density after remodeling around cementless femoral implants. Clinical Orthopaedics and Related Research 1994, 178.
94. Subbarayan G., Bartel D. A reconciliation of local and global models for bone remodeling through optimization theory. Journal of Biomechanical Engineering 2000, 122:72-76.
95. Sumner D., Turner T., Igloria R., Urban R., Galante J. Functional adaptation and ingrowth of bone vary as a function of hip implant stiffness. Journal of Biomechanics 1998, 31:909-917.
96. Tanck E., Ruimerman R., Huiskes R. Trabecular architecture can remain intact for both disuse and overload enhanced resorption characteristics. Journal of Biomechanics 2006, 39:2631-2637.
97. Taylor D., Casolari E., Bignardi C. Predicting stress fractures using a probabilistic model of damage, repair and adaptation. Journal of Orthopaedic Research 2004, 22:487-494.
98. Taylor D., Hazenberg J., Lee T. The cellular transducer in damage-stimulated bone remodelling: a theoretical investigation using fracture mechanics. Journal of Theoretical Biology 2003, 225:65-75.
99. Taylor D., Hazenberg J.G., Lee T.C. Living with cracks: damage and repair in human bone. Nature Materials 2007, 6:263-268.
100. Taylor D., Lee T. Microdamage and mechanical behaviour: predicting failure and remodelling in compact bone. Journal of Anatomy 2003, 203:203-211.
101. Tovar A. Bone remodeling as a hybrid cellular automaton optimization process 2004, University of Notre Dame.
102. Tovar A., Patel N.M., Niebur G.L., Sen M., Renaud J.E. Topology optimization using a hybrid cellular automaton method with local control rules. Journal of Mechanical Design 2006, 128:1205.
103. Trussel A., Muller R., Webster D. Toward Mechanical Systems Biology in Bone. Annals of Biomedical Engineering 2012, 1-13.
104. Tsubota K., Adachi T., Tomita Y. Functional adaptation of cancellous bone in human proximal femur predicted by trabecular surface remodeling simulation toward uniform stress state. Journal of Biomechanics 2002, 35:1541-1551.
105. Tsubota K., Suzuki Y., Yamada T., Hojo M., Makinouchi A., Adachi T. Computer simulation of trabecular remodeling in human proximal femur using large-scale voxel FE models: Approach to understanding Wolff's law. Journal of Biomechanics 2009, 42:1088-1094.
106. Turner A., Gillies R., Sekel R., Morris P., Bruce W., Walsh W. Computational bone remodelling simulations and comparisons with DEXA results. Journal of Orthopaedic Research 2005, 23:705-712.
107. Turner C., Chandran A., Pidaparti R. The anisotropy of osteonal bone and its ultrastructural implications. Bone 1995, 17:85-89.
108. Turner C.H., Anne V., Pidaparti R. A uniform strain criterion for trabecular bone adaptation: Do continuum-level strain gradients drive adaptation?. Journal of Biomechanics 1997, 30:555-563.
109. Turner C.H., Cowin S.C., Rho J.Y., Ashman R.B., Rice J.C. The fabric dependence of the orthotropic elastic constants of cancellous bone. Journal of Biomechanics 1990, 23:549-561.
110. Vahdati A., Rouhi G. A model for mechanical adaptation of trabecular bone incorporating cellular accommodation and effects of microdamage and disuse. Mechanics Research Communications 2009, 36:284-293.
111. Van der Helm F.C.T. A finite element musculoskeletal model of the shoulder mechanism. Journal of Biomechanics 1994, 27:551-569.
112. Van Rietbergen B., Odgaard A., Kabel J., Huiskes R. Relationships between bone morphology and bone elastic properties can be accurately quantified using high-resolution computer reconstructions. Journal of Orthopaedic Research 2005, 16:23-28.
113. Weinans H., Huiskes R., Grootenboer H. The behavior of adaptive bone-remodeling simulation models. Journal of Biomechanics 1992, 25:1425-1441.
114. Weinans H., Prendergast P. Tissue adaptation as a dynamical process far from equilibrium. Bone 1996, 19:143-150.
115. Weinans H., Sumner D.R., Igloria R., Natarajan R.N. Sensitivity of periprosthetic stress-shielding to load and the bone density-modulus relationship in subject-specific finite element models. Journal of Biomechanics 2000, 33:809-817.
116. Wolff J., Maquet P., Furlong R. The law of bone remodelling 1986, Springer-Verlag, Berlin.
117. Wong C., Gehrchen P.M., Kiaer T. Can experimental data in humans verify the finite element-based bone remodeling algorithm?. Spine 2008, 33:2875.
118. Zadpoor A.A., Campoli G., Weinans H. Neural network prediction of load from the morphology of trabecular bone. Applied Mathematical Modelling 2013, 37:5260-5276.
119. Zadpoor A.A., Nikooyan A.A. A mechanical model to determine the influence of masses and mass distribution on the impact force during running-A discussion. Journal of Biomechanics 2006, 39:388-390.
120. Zadpoor A.A., Nikooyan A.A. Modeling muscle activity to study the effects of footwear on the impact forces and vibrations of the human body during running. Journal of Biomechanics 2010, 43:186-193.
121. Zadpoor A.A., Nikooyan A.A., Arshi A.R. A model-based parametric study of impact force during running. Journal of Biomechanics 2007, 40:2012-2021.
122. Zysset P., Curnier A. A 3D damage model for trabecular bone based on fabric tensors. Journal of Biomechanics 1996, 29:1549-1558.