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Free surface density and microdamage in the bone remodeling equation: theoretical considerations. (English) Zbl 1213.74227
Summary: Bone is maintained through a coupled process of bone resorption and bone formation, in a continuous process called bone remodeling. An imbalance in this process caused by disease, abnormal mechanical demands, or fatigue may predispose bone to fracture injuries. The remodeling process is generally viewed as a material response to functional demands. Here, we propose a new set of constitutive equations for the bone remodeling process and contains the specific surface, instead of volume fraction, and the degree of microcracking in the constitutive equations. The rate of remodeling is related to mechanical stimuli, free surface density and a microcrack factor. In this approach, the effect of mechanical stimuli, rate of mechanical stimuli, and integration of mechanical stimuli on bone remodeling can be evaluated simultaneously in the remodeling equation. Specific examples are given for illustration of the model.

MSC:
74L15 Biomechanical solid mechanics
92C50 Medical applications (general)
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[1] Ascenzi, A., Biomechanics and galileo Galilei, J. biomech., 26, 95-100, (1993)
[2] Wolff, J.L., The law of bone remodeling, (1886), Springer Berlin
[3] Gjelsvik, A., Bone remodeling and piezoelectricity—I, J. biomech., 6, 69-77, (1973)
[4] Gjelsvik, A., Bone remodeling and piezoelectricity—II, J. biomech., 6, 187-193, (1973)
[5] Pollack, S.R.; Salzstein, R.; Pienkowski, D., The electric double layer in bone and its influence on stress-generated potentials, Calcif. tissue int., 36, Suppl. 1, S77-S81, (1984)
[6] Frost, H.M., Presence of microscopic cracks in vivo in bone, Bull. henry Ford hospital, 8, 25-35, (1960)
[7] Carter, D.R.; Hayes, W.C., Compact bone fatigue damage—I. residual strength and stiffness, J. biomech., 10, 325-337, (1977)
[8] Carter, D.R.; Cayler, W.E., Cycle dependent and time dependent bone fracture with repeated loading, J. biomech. eng., 105, 166-170, (1983)
[9] Carter, D.R.; Cayler, W.E., A cumulative damage model for bone fracture, J. orthop. res., 3, 84-90, (1985)
[10] Martin, R.B., A theory of fatigue damage accumulation and repair in cortical bone, J. orthop. res., 10, 818-825, (1992)
[11] Martin, R.B., Mathematical model for repair of fatigue damage and stress fracture in osteonal bone, J. orthop. res., 13, 309-316, (1995)
[12] Prendergast, P.J.; Taylor, D., Prediction of bone adaptation using damage accumulation, J. biomech., 27, 1067-1076, (1994)
[13] Cowin, S.C.; Moss Salentijn, L.; Moss, M.L., Candidates for the mechanosensory system in bone, J. biomech. eng., 113, 191-197, (1991)
[14] Cowin, S.C.; Weinbaum, S.; Zeng, Y., A case for bone canaliculi as the anatomical site of strain generated potentials, J. biomech., 28, 1281-1297, (1995)
[15] Carter, D.R., Mechanical loading history and skeletal biology, J. biomech., 20, 1095-1109, (1987)
[16] Carter, D.R.; Van der Meulen, M.C.; Beaupre, G.S., Mechanical factors in bone growth and development, Bone, 18, Suppl. 1, S5-S10, (1996)
[17] Martin, R.B., Geometric feedback in bone remodeling and osteoporosis, J. biomech., 5, 447-455, (1972)
[18] R.B. Martin, The specific surface – porosity relationship in bone: refinement of the concept of geometric feedback, in: Orthopaedic Research Society Meeting, Dallas, 14-17 January 1974.
[19] Martin, R.B., Porosity and specific surface of bone, Critical reviews in biomedical engineering, (1984), CRC Boca Raton, FL, pp. 179-222
[20] Cowin, S.C.; Hegedus, D.D., Bone remodeling I: A theory of adaptive elasticity, J. elasticity, 6, 313-325, (1976) · Zbl 0335.73028
[21] Hegedus, D.D.; Cowin, S.C., Bone remodeling II: small strain adaptive elasticity, J. elasticity, 6, 337-352, (1976) · Zbl 0342.73069
[22] Cowin, S.C.; Van Buskirk, W.C., Internal bone remodeling induced by a medullary pin, J. biomech., 11, 5, 269-275, (1978)
[23] Cowin, S.C.; Van Buskirk, W.C., Surface bone remodeling induced by a medullary pin, J. biomech., 12, 4, 269-276, (1979)
[24] Cowin, S.C.; Firoozbakhsh, K., Bone remodeling of diaphysial surfaces under constant load: theoretical predictions, J. biomech., 7, 471-484, (1981)
[25] Firoozbakhsh, K.; Cowin, S.C., An analytical model of pauwels’ functional adaptation mechanism in bone, J. biomech. eng. (ASME), 103, 246-252, (1981)
[26] Beaupre, G.S.; Orr, T.E.; Carter, D.R., An approach for time-dependent bone modeling and remodeling—theoretical development, J. orthop. res., 8, 651-661, (1990)
[27] Beaupre, G.S.; Orr, T.E.; Carter, D.R., An approach for time-dependent bone modeling and remodeling—a preliminary remodeling simulation, J. orthop. res., 8, 662-670, (1990)
[28] Jacobs, C.R.; Simo, J.C.; Beaupre, G.S.; Carter, D.R., Adaptive bone remodeling incorporating simultaneous density and anisotropy considerations, J. biomech., 30, 6, 603-613, (1997)
[29] Mullender, M.G.; Huiskes, R.; Weinans, H., A physiological approach to the simulation of bone remodeling as a self-organizational control process, J. biomech., 27, 11, 1389-1394, (1994)
[30] Prendergast, P.J.; Huiskes, R., Microdamage and osteocyte-lacuna strain in bone. A micro structural finite element analysis, J. biomech. eng., 118, 240-246, (1996)
[31] Zidi, M.; Ramtani, S., Bone remodeling theory applied to the study of n unit elements model, J. biomech., 32, 743-747, (1999) · Zbl 0978.74053
[32] Ramtani, S.; Zidi, M., Damaged-bone remodeling theory: thermodynamical approach, Mech. res. commun., 26, 2, 701-708, (1999) · Zbl 0978.74053
[33] Ramtani, S.; Zidi, M., Remodeling of the bone material containing microcracks: a theoretical analysis, Eur. phys. J., AP, 8, 257-263, (1999) · Zbl 0978.74053
[34] Ramtani, S.; Zidi, M., A theoretical model of the effect of continuum damage on a bone adaptation model, J. biomech., 34, 471-479, (2001)
[35] Papathanasopoulou, V.A.; Fotiadis, D.I.; Foutsitzi, G.; Massalas, C.V., A poroelastic bone model for internal remodeling, Int. J. eng. sci., 40, 5, 511-530, (2002)
[36] Papathanasopoulou, V.A.; Fotiadis, D.I.; Massalas, C.V., A theoretical analysis of surface remodeling in long bones, Int. J. eng. sci., 42, 395-409, (2004)
[37] Rouhi, G.; Herzog, W.; Sudak, L.; Firoozbakhsh, K.; Epstein, M., Free surface density instead of volume fraction in the bone remodeling equation: theoretical considerations, Forma, 19, 3, 165-182, (2004) · Zbl 1200.74110
[38] Eriksen, E.F.; Kassem, M., The cellular basis of bone remodeling, Triangle, 31, 2/3, 45-57, (1992)
[39] Carter, D.R.; Beaupre, G.S., Skeletal function and form, mechanobiology of skeletal development, aging and regeneration, (2001), Cambridge University Press NY, USA
[40] T.C. Lee, Detection and accumulation of microdamage in bone, MD thesis, University of Dublin, Ireland, 1997.
[41] Burr, D.B.; Milgrom, C.; Boyd, R.D.; Higgins, W.L.; Radin, E.L., Experimental stress fractures of the tibia-biological and mechanical etiology in rabbits, J. bone joint surg., 72B, 370-375, (1990)
[42] Burr, D.B.; Martin, R.B.; Schaffler, M.B.; Radin, E.L.; Milgrom, C., Bone remodeling in response to in vivo fatigue micro-damage, J. biomech., 18, 189-200, (1985)
[43] Mori, S.; Burr, D.B., Increased intracortical remodeling following fatigue damage, Bone, 14, 103-109, (1993)
[44] Martin, R.B., Is all cortical bone remodeling initiated by microdamage?, Bone, 30, 8-13, (2002)
[45] Martin, R.B., Fatigue microdamage as an essential element of bone mechanics and biology, Calcif tissue int., 73, 101-107, (2003)
[46] Taylor, D.; Hazenberg, J.G.; Lee, T.C., The cellular transducer in damage-stimulated bone remodeling: a theoretical investigation using fracture mechanics, J. theor. biol., 225, 1, 65-75, (2003)
[47] Hernandez, C.J.; Beaupré, G.S.; Carter, D.R., A model of mechanobiologic and metabolic influences on bone adaptation, J. reh. res. dev. (JRRD), 37, 2, 235-244, (2000)
[48] Huiskes, R., Simulation of self-organization and functional adaptation in bone, (1997), Springer Berlin, pp. 299-319
[49] Burr, D.B.; Martin, R.B., Calculating the probability that microcracks initiate resorption spaces, J. biomech., 26, 613-616, (1993)
[50] Coleman, B.D.; Gurtin, M., Thermodynamics with internal variables, J. chem. phys., 47, 597-613, (1967)
[51] Miehe, C., Discontinuous and continuous damage evolution in ogden-type large strain elastic materials, Euro. J. mech., A/solids, 14, 5, 697-720, (1995) · Zbl 0837.73054
[52] R.B. Martin, 2005, Personal communications.
[53] Fritton, S.P.; McLeod, K.J.; Rubin, C.T., Quantifying the strain history of bone: spatial uniformity and self similarity of low magnitude strains, J. biomech., 33, 3, 317-325, (2000)
[54] Nicolella, D.P.; Nicholls, A.E.; Lankford, J.; Davy, D.T., Machine vision photogrammetry: a technique for measurement of microstructural strain in cortical bone, J. biomech., 34, 135-139, (2001)
[55] O’Brien, F.J.; Taylor, D.; Dickson, G.R.; Lee, T.C., Visualization of three-dimensional microcracks in compact bone, J. anat., 197, 413-420, (2000)
[56] Taylor, D., Modeling of fatigue crack growth at the micro-structural level, Comp. mat. sci., 25, 1-2, 228-236, (2002)
[57] Harrigan, T.P.; Jasty, M.; Mann, R.W.; Harris, W.H., Limitations of the continuum assumption in cancellous bone, J. biomech., 21, 4, 269-275, (1988)
[58] Chamay, A.; Tschantz, P., Mechanical influences in bone remodeling: experimental research on wolff’s law, J. biomech., 5, 2, 173-180, (1972)
[59] Rubin, C.T.; McLeod, K.J., Promotion of bony ingrowth by frequency specific, low amplitude mechanical strains, Clin. orth. rel. res., 298, 165-174, (1994)
[60] Forwood, M.R.; Turner, C.H., Skeletal adaptations to mechanical usage: results from tibial loading studies in rats, Bone, 17, Suppl. 4, S197-S205, (1995)
[61] Mosley, J.R.; Lanyon, L.E., Strain rate as a controlling influence on adaptive modeling in response to dynamic loading of the ulna in growing male rats, Bone, 23, 4, 313-318, (1998)
[62] Qin, Y.X.; Rubin, C.T.; Mcleod, K.J., Nonlinear dependence of loading intensity and cycle number in the maintenance of bone mass and morphology, J. orthop. res., 16, 4, 482-489, (1998)
[63] Turner, C.H., Three rules for bone adaptation to mechanical stimuli, Bone, 23, 5, 549-561, (1998)
[64] Ward, K.; Alsop, C.; Brown, S.; Caulton, J.; Rubin, C.T.; Adams, J.; Mughal, M., Low magnitude, high frequency loading therapy increases volumetric tibial bone mineral density in children with disabling conditions, J. bone mineral res., 19, 360-369, (2004)
[65] Carter, D.R., Mechanical loading histories and cortical bone remodeling, Calcif. tissue int., 36, Suppl. 1, S19-S24, (1984)
[66] Lanyon, L.E.; Rubin, C.T., Static vs dynamic loads as an influence on bone remodeling, J. biomech., 17, 12, 897-905, (1984)
[67] Carter, D.R.; Orr, T.E.; Fyhrie, D.P., Relationship between loading history and femoral cancellous bone architecture, J. biomech., 22, 3, 231-244, (1989)
[68] Carter, D.R.; Hayes, W.C., The compressive behavior of bone as a two-phase porous structure, J. bone joint surg., 59A, 954-962, (1977)
[69] Gibson, L.J., The mechanical behavior of cancellous bone, J. biomech., 18, 317-328, (1985)
[70] Orr, T.E.; Beaupre, G.S.; Carter, D.R.; Schurman, D.J., Computer predictions of bone remodeling around porous-coated implants, J. arthop., 5, 191-200, (1990)
[71] Huiskes, R.; Weinans, H.; Grootenboer, H.J.; Dalstra, M.; Fudala, B.; Sloof, T.J., Adaptive bone remodeling theory applied to prosthetic-design analysis, J. biomech., 20, 11-12, 167-182, (1987)
[72] Van Rietbergen, B.; Huiskes, R.; Weinans, H.; Sumner, D.R.; Turner, T.M.; Galante, J.O., The mechanism of bone remodeling and resorption around press-fitted THA stems, J. biomech., 26, 4/5, 369-382, (1993)
[73] Cowin, S.C., Adaptive elasticity: a review and critique of a bone tissue adaptation model, Eng. trans. Polish acad. sci., 51, 2-3, 113-193, (2003) · Zbl 1064.74129
[74] ()
[75] ()
[76] Weinbaum, S.; Cowin, S.C.; Zeng, Y., A model for the excitation of osteocytes by mechanical loading-induced bone fluid shear stresses, J. biomech., 27, 339-360, (1994)
[77] Cowin, S.C.; Moss, M.L., Mechanosensory mechanisms in bone, (), 723-738
[78] Rubin, C.; Turner, A.S.; Bain, S.; Mallinckrodt, C.; McLeod, C., Anabolism: low mechanical signals strengthen long bones, Nature, 412, 603-604, (2002)
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