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Multiscale modeling of elastic properties of cortical bone. (English) Zbl 1320.74078

Summary: We model cortical bone as a composite material with hierarchical structure. At a nanostructural level, bone is composed of cross-linked collagen molecules, containing water and non-collagenous proteins in their gaps, reinforced with hydroxyapatite-like nanocrystals. Such a nanocomposite structure represents a mineralized collagen fibril, which serves as a primary building block of bone. At a sub-microstructural level (few microns), the mineralized collagen fibrils are embedded in an extrafibrillar hydroxyapatite matrix to form a single lamella, which also contains the lacunar cavities. At a microstructural level (hundreds of microns) one can distinguish two lamellar structures in the mature cortical bone: osteons, made of concentric layers of lamellae surrounding long hollow Haversian canals, and interstitial lamellae made of remnants of old osteons. At a mesostructural level (several millimeters), the cortical bone is represented by a random collection of osteons and resorption cavities in the interstitial lamellae. A macrostructural level is the whole bone level containing both the cortical (compact) and trabecular (spongy) bone types. In this paper, we predict analytically the effective elastic constants of cortical bone by modeling its elastic response at these different scales, spanning from the nanostructural to mesostructural levels, using micromechanics methods and composite materials laminate theories. The results obtained at a lower scale serve as inputs for the modeling at a higher scale. The predictions are in good agreement with the experimental data reported in literature.

MSC:

74L15 Biomechanical solid mechanics
92C10 Biomechanics
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[1] Landis W.J.: The strength of a calcified tissue depends in part on the molecular-structure and organization of its constituent mineral crystals in their organic matrix. Bone 16, 533–544 (1995)
[2] Weiner S., Wagner H.D.: Material bone: structure-mechanical function relations. Annu. Rev. Mater. Sci. 28, 271–298 (1998)
[3] Feng, L.: Multi-scale characterization of swine femoral cortical bone and long bone defect repair by regeneration. Ph.D. dissertation, University of Illinois at Urbana-Champaign (2010)
[4] Olszta M.J., Cheng X., Jee S.S., Kumar R., Kim Y., Kaufman M.J., Douglas E.P., Gower L.B.: Bone structure and formation: a new perspective. Mater. Sci. Eng. R Rep. 58, 77–116 (2007)
[5] Robinson R.: An electron microscopic study of the crystalline inorganic component of bone and its relationship to the organic matrix. J. Bone Joint Surg. 34, 389–435 (1952)
[6] Rho J., Kuhn-Spearing L., Zioupos P.: Mechanical properties and the hierarchical structure of bone. Med. Eng. Phys. 20, 92–102 (1998)
[7] Buehler M.J.: Nanomechanics of collagen fibrils under varying cross-link densities: atomistic and continuum studies. J. Mech. Behav. Biomed. Mater. 1, 59–67 (2008)
[8] Currey J.D.: The relationship between the stiffness and the mineral content of bone. J. Biomech. 2, 477–480 (1969)
[9] Katz J.L.: Hard tissue as a composite material-I. Bounds on the elastic behavior. J. Biomech. 2, 455–473 (1971)
[10] Halpin J.C., Kardos J.L.: Halpin-Tsai equations: a review. Polym. Eng. Sci. 16, 344–352 (1976)
[11] Hirsch T.J.: Modulus of elasticity of concrete affected by elastic moduli of cement paste matrix and ggregate. ACI J. 59, 427–451 (1962)
[12] Tandon G.P., Weng G.J.: Effect of aspect ratio of inclusions on the elastic properties of unidirectionally aligned composites. Polym. Compos. 5, 327–333 (1983)
[13] Akkus O.: Elastic deformation of mineralized collagen fibrils: an equivalent inclusion based composite model. J. Biomech. Eng. 127, 383–390 (2005)
[14] Fritsch A., Hellmich C.: Universal microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: micromechanics-based prediction of anisotropic elasticity. J. Theor. Biol. 244, 597–620 (2007)
[15] Nikolov S., Raabe D.: Hierarchical modeling of the elastic properties of bone at submicron scales: the role of extrafibrillar mineralization. Biophys. J. 94, 4220–4232 (2008)
[16] Yoon Y.J., Cowin S.C.: The estimated elastic constants for a single bone osteonal lamella. Biomech. Model. Mechanobiol. 7, 1–11 (2008)
[17] Ji B., Gao H.: Elastic properties of nanocomposite structure of bone. Compos. Sci. Technol. 66, 1209–1215 (2006)
[18] Kotha S.P., Guzelsu N.: The effects of interphase and bonding on the elastic modulus of bone: changes with age-related osteoporosis. Med. Eng. Phys. 22, 575–585 (2000)
[19] Siegmund T., Allen M.R., Burr D.B.: Failure of mineralized collagen fibrils: modeling the role of collagen cross-linking. J. Biomech. 41, 1427–1435 (2008)
[20] Jasiuk I., Ostoja-Starzewski M.: Modeling of bone at a single lamella level. Biomech. Model. Mechanobiol. 3, 67–74 (2004)
[21] Dong X.N., Guo X.E.: Prediction of cortical bone elastic constants by a two-level micromechanical model using a generalized self-consistent method. J. Biomech. Eng. 128, 309–316 (2006)
[22] Lees S.: Considerations regarding the structure of the mammalian mineralized osteoid from viewpoint of the generalized packing model. Connect. Tissue Res. 16, 281–303 (1987)
[23] Bailey A.J., Paul R.G.: Mechanisms and consequences of the maturation and ageing of collagen. Proc. Indian Acad. Sci. Chem. Sci. 111, 57–69 (1999)
[24] Lees S., Pineri M., Escoubes M.: A generalized packing model for type I collagen. Int. J. Biol. Macromol. 6, 133–136 (1984)
[25] Fratzl P., Gupta H.S., Paschalis E.P., Roschger P.: Structure and mechanical quality of the collagen–mineral nano-composite in bone. J. Mater. Chem. 14, 2115–2123 (2004)
[26] Katz E.P., Li S.T.: Structure and function of bone collagen fibrils. J. Mol. Biol. 80, 1–15 (1973)
[27] Sasaki N., Sudoh Y.: X-ray pole figure analysis of apatite crystals and collagen molecules in bone. Calcif. Tissue Int. 60, 361–367 (1997)
[28] Lees S., Prostak K.S., Ingle V.K., Kjoller K.: The loci of mineral in turkey leg tendon as seen by atomic force microscope and electron microscopy. Calcif. Tissue Int. 55, 180–189 (1994)
[29] Sasaki N., Tagami A., Goto T., Taniguchi M., Nakata M., Hikichi K.: Atomic force microscopic studies on the structure of bovine femoral cortical bone at the collagen fibril-mineral level. J. Mater. Sci. Mater. Med. 13, 333–337 (2002)
[30] Mori T., Tanaka K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica 21, 571–574 (1973)
[31] Benveniste Y.: New approach to the application of Mori–Tanaka’s theory in composite materials. Mech. Mater. 6, 147–157 (1987)
[32] Eshelby J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 241, 376–396 (1957) · Zbl 0079.39606
[33] Gavazzi A.C., Lagoudas D.C.: On the numerical evaluation of Eshelby’s tensor and its application to elastoplastic fibrous composites. Comput. Mech. 7, 13–19 (1990)
[34] Hassenkam T., Fantner G.E., Cutroni J.A., Weaver J.C., Morse D.E., Hansma P.K.: High-resolution AFM imaging of intact and fractured trabecular bone. Bone 35, 4–10 (2004)
[35] Zhang W., Liao S.S., Cui F.Z.: Hierarchical self-assembly of nano-fibrils in mineralized collagen. Chem. Mater. 15, 3221–3226 (2003)
[36] Fratzl P., Schreiber S., Klaushofer K.: Bone mineralization as studied by small-angle x-ray scattering. Connect. Tissue Res. 34, 247–254 (1996)
[37] Benezra R., Hobbs L.W., Spector M.: The ultrastructure of anorganic bovine bone and selected synthetic hyroxyapatites used as bone graft substitute materials. Biomaterials 23, 921–928 (2002)
[38] Hellmich C., Barthelemy J., Dormieux L.: Mineral-collagen interactions in elasticity of bone ultrastructure–a continuum micromechanics approach. Eur. J. Mech. A/Solids 23, 783–810 (2004) · Zbl 1058.74584
[39] Fritsch A., DormieuxL. Hellmich C.: Porous polycrystals built up by uniformly and axisymmetrically oriented needles: homogenization of elastic properties. C. R. Mech. 334, 151–157 (2006) · Zbl 1372.74015
[40] Fritsch A., Hellmich C., Dormieux L.: Ductile sliding between mineral crystals followed by rupture of collagen crosslinks: experimentally supported micromechanical explanation of bone strength. J. Theor. Biol. 260, 230–252 (2009) · Zbl 1402.92035
[41] Hill R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11, 357–372 (1963) · Zbl 0114.15804
[42] Budiansky B.: On the elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids 13, 223–227 (1965)
[43] Hellmich C., Ulm F.: Micromechanical model for ultrastructural stiffness of mineralized tissues. J. Eng. Mech. 128, 898–908 (2002)
[44] Cowin S.C.: Bone Mechanics Handbook. CRC Press, Boca Raton (2001)
[45] Giraud-Guille M.M.: Twisted plywood architecture of collagen fibrils in human compact bone osteons. Calcif. Tissue Int. 42, 167–180 (1988)
[46] Martin R.B., Burr D.B., Sharkey N.A.: Skeletal Tissue Mechanics. Springer, New York (1998)
[47] Sun C.T., Li S.: Three-dimensional effective elastic constants for thick laminates. J. Compos. Mater. 22, 629–639 (1988)
[48] Daniel I.M., Ishai O.: Engineering Mechanics of Composite Materials. Oxford University Press, New York (2006)
[49] Burr D.B., Schaffler M.B., Frederickson R.G.: Composition of the cement line and its possible mechanical role as a local interface in human compact bone. J. Biomech. 21, 939–945 (1988)
[50] Guo X.E., Liang L.C., Goldstein S.A.: Micromechanics of osteonal cortical bone fracture. J. Biomech. Eng. 120, 112–117 (1998)
[51] Taya M., Chou T.W.: On two kinds of ellipsoidal inhomogeneities in an infinite elastic body: an application to a hybrid composite. Int. J. Solids Struct. 17, 553–563 (1981) · Zbl 0476.73052
[52] Mura T.: Micromechanics of Defects in Solids. Martinus Nijhoff, The Hague (1982)
[53] Buehler M.J.: Atomistic and continuum modeling of mechanical properties of collagen: elasticity, fracture, and self-assembly. J. Mater. Res. 21, 1947–1961 (2006)
[54] Veld P.J., Stevens M.J.: Simulation of the mechanical strength of a single collagen molecule. Biophys. J. 95, 33–39 (2008)
[55] Boskey A.L.: Bone mineralization: age and sex differences. In: Cowin, S.C. (eds) Bone Mechanics Handbook, CRC Press, Boca Raton (2001)
[56] Snyders R., Music D., Sigumonrong D., Schelnberger B., Jensen J., Schneider J.M.: Experimental and ab initio study of the mechanical properties of hydroxyapatite. Appl. Phys. Lett. 90(193902), 1–13 (2007)
[57] Gupta H.S., Seto J., Wagermaier W., Zaslansky P., Boesecke P., Fratzl P.: Cooperative deformation of mineral and collagen in bone at the nanoscale. Proc. Natl. Acad. Sci. USA 103, 17741–17746 (2006)
[58] Remaggi F., CaneI V., Palumbo C., Ferretti M.: Histomorphometric study on the osteocyte lacuno-canalicular network in animals of different species. I. Woven-fibered and parallel-fibered bones. Ital. J. Anat. Embryol. 103, 145–155 (1998)
[59] Rho J.Y., Zioupos P., Currey J.D., Pharr G.M.: Variations in the individual thick lamellar properties within osteons by nanoindentation. Bone 25, 295–300 (1999)
[60] Cheng L., Wang L., Karlsson A.M.: Image analyses of two crustacean exoskeletons and implications of the exoskeletal microstructure on the mechanical behavior. J. Mater. Res. 23, 2854–2872 (2008)
[61] Grant, C.A., Brockwell, D.J., Radford, S.E., Thomson, N.H.: Effects of hydration on the mechanical response of individual collagen fibrils. Appl. Phys. Lett. 92 (2008)
[62] Harley R., James D., Miller A., White J.W.: Phonons and the elastic moduli of collagen and muscle. Nature 267, 285–287 (1977)
[63] Cusack S., Miller A.: Determination of the elastic constants of collagen by Brillouin light scattering. J. Mol. Biol. 135, 39–51 (1979)
[64] Van D.R., Van D.W., Bennink M.L., Dijkstra P.J., Feijen J.: Micromechanical testing of individual collagen fibrils. Macromol. Biosci. 6, 699–702 (2006)
[65] Yang L., Van Der Werf K.O., Koopman B.F., Subramaniam V., Bennink M.L., Dijkstra P.J., Feijen J.: Micromechanical bending of single collagen fibrils using atomic force microscopy. J. Biomed. Mater. Res. A 82, 160–168 (2007)
[66] Buehler M.J.: Molecular nanomechanics of nascent bone: fibrillar toughening by mineralization. Nanotechnology 18, 295102–295111 (2007)
[67] Bhowmik R., Katti K.S., Katti D.R.: Mechanics of molecular collagen is influenced by hydroxyapatite in natural bone. J. Mater. Sci. 42, 8795–8803 (2007)
[68] Dubey D.K., Tomar V.: Microstructure dependent dynamic fracture analyses of trabecular bone based on nascent bone atomistic simulations. Mech. Res. Commun. 35, 24–31 (2008) · Zbl 1258.74153
[69] Erts D., Gathercole L.J., Atkins E.D.T.: Scanning probe microscopy of intrafibrillar crystallites in calcified collagen. J. Mater. Sci. Mater. Med. 5, 200–206 (1994)
[70] Rubin M.A., Jasiuk I., Taylor J., Rubin J., Ganey T., Apkarian R.P.: TEM analysis of the nanostructure of normal and osteoporotic human trabecular bone. Bone 33, 270–282 (2003)
[71] Weiner S., Traub W.: Bone structure: from angstroms to microns. FASEB J. 6, 879–885 (1992)
[72] Glimcher M.J.: A basic architectural principle in the organization of mineralized tissues. Clin. Orthop. 61, 16–36 (1968)
[73] Glimcher M.J.: Recent studies of the mineral phase in bone and its possible linkage to the organic matrix by protein-bound phosphate bonds. Philos. Trans. R. Soc. Lond. 304, 479–508 (1984)
[74] Lee D.D., Glimcher M.J.: The three-dimensional spatial relationship between the collagen fibrils and the inorganic calcium-phosphate crystals of pickerel and herring fish bone. Connect. Tissue Res. 21, 247–257 (1989)
[75] Rubin M.A., Jasiuk I.: The TEM characterization of the lamellar structure of osteoporotic human trabecular bone. Micron 36, 653–664 (2005)
[76] Turner C.H, Chandran A., Pidaparti R.M.V.: The anisotropy of osteonal bone and its ultrastruetural implications. Bone 17, 85–89 (1995)
[77] Giraud-Guille M., Besseau L., Martin R.: Liquid crystalline assemblies of collagen in bone and in vitro systems. J. Biomech. 36, 1571–1579 (2003)
[78] Weiner S., Arad T., Sabanay I., Traub W.: Rotated plywood structure of primary lamellar bone in the rat: orientations of the collagen fibril arrays. Bone 20, 509–514 (1997)
[79] Timlin J.A., Carden A., Morris M.D.: Chemical microstructure of cortical bone probed by Raman transects. Appl. Spectrosc. 53, 1429–1435 (1999)
[80] Boivin G., Meunier P.J.: The degree of mineralization of bone tissue measured by computerized quantitative contact microradiography. Calcif. Tissue Int. 70, 503–511 (2002)
[81] Bonfield W., Li C.H.: Anisotropy of nonelastic flow in bone. J. Appl. Phys. 38, 2450–2455 (1967)
[82] Katz J.L., Meunier A.: The elastic anisotropy of bone. J. Biomech. 20, 1063–1070 (1987)
[83] Gilmore R.S., Katz J.L.: Elastic properties of apatites. J. Mater. Sci. 17, 1131–1141 (1982)
[84] Hofmann H., Voss T., Kühn K., Engel J.: Localization of flexible sites in thread-like molecules from electron micrographs. Comparison of interstitial, basement membrane and intima collagens. J. Mol. Biol. 172, 325–343 (1984)
[85] Mammone J.F., Hudson S.M.: Micromechanics of bone strength and fracture. J. Biomech. 26, 439–446 (1993)
[86] Pidaparti R.M.V., Chandran A., Takano Y., Turner C.H.: Bone mineral lies mainly outside collagen fibrils: Predictions of a composite model for osteonal bone. J. Biomech. 29, 909–916 (1996)
[87] Vesentini S., Fitie F.C., Montevecchi F.M., Redaelli A.: Molecular assessment of the elastic properties of collagen-like homotrimer sequences. Biomech. Model. Mechanobiol. 3, 224–234 (2005)
[88] Balooch M., Habelitz S., Kinney J.H., Marshall S.J., Marshall G.W.: Mechanical properties of mineralized collagen fibrils as influenced by demineralization. J. Struct. Biol. 162, 404–410 (2008)
[89] Minary-Jolandan M., Yu M.: Nanomechanical heterogeneity in the gap and overlap regions of type I collagen fibrils with implications for bone heterogeneity. Biomacromolecules 10, 2565–2570 (2009)
[90] Sasaki N., Ikawa T., Fukuda A.: Orientation of mineral in bovine bone and the anisotropic mechanical properties of plexiform bone. J. Biomech. 24, 57–61 (1991)
[91] Wagner H.D., Weiner S.: On the relationship between the microstructure of bone and its mechanical stiffness. J. Biomech. 25, 1311–1320 (1992)
[92] Jager I., Fratzl P.: Mineralized collagen fibrils: a mechanical model with a staggered arrangement of mineral particles. Biophys. J. 79, 1737–1746 (2000)
[93] Rho J., Tsui T.Y., Pharr G.M.: Elastic properties of human cortical and trabecular lamellar bone measured by nanoindentation. Biomaterials 18, 1325–1330 (1997)
[94] Zysset P.K., Guo X.E., Hoffler C.E., Moore K.E., Goldstein S.A.: Elastic modulus and hardness of cortical and trabecular bone lamellae measured by nanoindentation in the human femur. J. Biomech. 32, 1005–1012 (1999)
[95] Hoffler C.E., Moore K.E., Kozloff K., Zysset P.K., Goldstein S.A.: Age, gender, and bone lamellae elastic moduli. J. Orthop. Res. 18, 432–437 (2000)
[96] Goodwin K.J., Sharkey N.A.: Material properties of interstitial lamellae reflect local strain environments. J. Orthop. Res. 20, 600–606 (2002)
[97] Raum K., Cleveland R.O., Peyrin F., Laugier P.: Derivation of elastic stiffness from site-matched mineral density and acoustic impedance maps. Phys. Med. Biol. 51, 747–758 (2006)
[98] Ascenzi A., Bonucci E.: The tensile properties of single osteons. Anat. Rec. 158, 375–386 (1967)
[99] Tho, H.B., Stolz, C., Vanleene, M., Bensamoun, S., Treutenaere, J., Rey, C.: Multi-scale characterization and modelling of human cortical bone. In: MRS Fall Meeting 28 Nov 2005–2 Dec 2005, pp. 60–65
[100] Yoon H.S., Katz J.L.: Ultrasonic wave propagation in human cortical bone. II. Measurements of elastic properties and microhardness. J. Biomech. 9, 459–464 (1976)
[101] Rho J.Y., Ashman R.B., Turner H.: Young’s modulus of trabecular and cortical bone material: ultrasonic and microtensile measurements. J. Biomech. 26, 111–119 (1993)
[102] Turner C.H., Rho J., Takano Y., Tsui T.Y., Pharr G.M.: The elastic properties of trabecular and cortical bone tissues are similar: results from two microscopic measurement techniques. J. Biomech. 32, 437–441 (1999)
[103] Cuppone M., Seedhom B.B., Berry E., Ostell A.E.: The longitudinal Young’s modulus of cortical bone in the midshaft of human femur and its correlation with CT scanning data. Calcif. Tissue Int. 74, 302–309 (2004)
[104] Dong X.N., Guo X.E.: The dependence of transversely isotropic elasticity of human femoral cortical bone on porosity. J. Biomech. 37, 1281–1287 (2004)
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