Elastic deformation of mineralized collagen fibrils: an equivalent inclusion based composite model.

Mineralized collagen fibrils are the basic building blocks of bone tissue at the supramolecular level. Several disease states, manipulation of the expression of specific proteins involved in biomineralization, and treatment with different agents alter the extent of mineralization as well as the morphology of mineral crystals which in turn affect the mechanical function of bone tissue. An experimental assessment of mineralized fibers' mechanical properties is challenged by their small size, leaving analytical and computational models as a viable alternative for investigation of the fibril-level mechanical properties. In the current study the variation of the elastic stiffness tensor of mineralized collagen fibrils with changing mineral volume fraction and mineral aspect ratios was predicted via a micromechanical model. The partitioning of applied stresses between mineral and collagen phases is also predicted for normal and shear loading of fibrils. Model predictions resulted in transversely isotropic collagen fibrils in which the modulus along the longer axis of the fibril was the greatest. All the elastic moduli increased with increasing mineral volume fraction whereas Poisson's ratios decreased with the exception of v12 (=v21). The partitioning of applied stresses were such that the stresses acting on mineral crystals were about 1.5, 15, and 3 times greater than collagen stresses when fibrils were loaded transversely, longitudinally, and in shear, respectively. In the overall the predictions were such that: (a) greatest modulus along longer axis; (b) the greatest mineral/collagen stress ratio along the longer axis of collagen fibers (i.e., greatest relief of stresses acting on collagen); and (c) minimal lateral contraction when fibers are loaded along the longer axis. Overall, the pattern of mineralization as put forth in this model predicts a superior mechanical function along the longer axis of collagen fibers, the direction which is more likely to experience greater stresses.

[1]  N. Sasaki,et al.  Stress-strain curve and Young's modulus of a collagen molecule as determined by the X-ray diffraction technique. , 1996, Journal of biomechanics.

[2]  C. M. Agrawal,et al.  Age-related changes in the collagen network and toughness of bone. , 2002, Bone.

[3]  J. Petruska,et al.  A SUBUNIT MODEL FOR THE TROPOCOLLAGEN MACROMOLECULE. , 1964, Proceedings of the National Academy of Sciences of the United States of America.

[4]  A. Boskey,et al.  The effects of noncollagenous matrix proteins on hydroxyapatite formation and proliferation in a collagen gel system. , 1989, Connective tissue research.

[5]  J. Katz.  Hard tissue as a composite material. I. Bounds on the elastic behavior. , 1971, Journal of biomechanics.

[6]  S. Weiner,et al.  Growth of mineral crystals in turkey tendon collagen fibers. , 1992, Connective tissue research.

[7]  T. Tait,et al.  A clinical and biochemical assessment of methotrexate in rheumatoid arthritis , 1994, Clinical Rheumatology.

[8]  M. Marko,et al.  Mineralization of collagen may occur on fibril surfaces: evidence from conventional and high-voltage electron microscopy and three-dimensional imaging. , 1996, Journal of structural biology.

[9]  P. Fratzl,et al.  Effects of sodium fluoride and alendronate on the bone mineral in minipigs: A small‐angle X‐ray scattering and backscattered electron imaging study , 1996, Journal of bone and mineral research : the official journal of the American Society for Bone and Mineral Research.

[10]  M. Glimcher,et al.  Isolation of calcium‐phosphate crystals of bone by non‐aqueous methods at low temperature , 1995, Journal of bone and mineral research : the official journal of the American Society for Bone and Mineral Research.

[11]  C. Rey,et al.  Fourier transform infrared spectroscopic study of the carbonate ions in bone mineral during aging , 1991, Calcified Tissue International.

[12]  P. Zysset,et al.  A combined atomic force microscopy and nanoindentation technique to investigate the elastic properties of bone structural units. , 2001, European cells & materials.

[13]  E. Atkins,et al.  Scanning probe microscopy of intrafibrillar crystallites in calcified collagen , 1994 .

[14]  H. Berendsen,et al.  Hydration structure of collagen and influence of salts. , 1966, Federation proceedings.

[15]  S. Weiner,et al.  On the relationship between the microstructure of bone and its mechanical stiffness. , 1992, Journal of biomechanics.

[16]  W. Landis.  The strength of a calcified tissue depends in part on the molecular structure and organization of its constituent mineral crystals in their organic matrix. , 1995, Bone.

[17]  S Lees,et al.  The locus of mineral crystallites in bone. , 1988, Connective tissue research.

[18]  D B Burr,et al.  Elastic anisotropy and collagen orientation of osteonal bone are dependent on the mechanical strain distribution , 1999, Journal of orthopaedic research : official publication of the Orthopaedic Research Society.

[19]  P. Fratzl,et al.  Mineralized collagen fibrils: a mechanical model with a staggered arrangement of mineral particles. , 2000, Biophysical journal.

[20]  R. Martin,et al.  The relative effects of collagen fiber orientation, porosity, density, and mineralization on bone strength. , 1989, Journal of biomechanics.

[21]  P. Withers,et al.  An introduction to metal matrix composites , 1993 .

[22]  N. Sasaki,et al.  Measurement of partition of stress between mineral and collagen phases in bone using X-ray diffraction techniques. , 1997, Journal of biomechanics.

[23]  V. A. Gibson,et al.  Collagen fiber organization is related to mechanical properties and remodeling in equine bone. A comparison of two methods. , 1996, Journal of biomechanics.

[24]  V. Ingle,et al.  The loci of mineral in turkey leg tendon as seen by atomic force microscope and electron microscopy , 1994, Calcified Tissue International.

[25]  S. Weiner,et al.  Ultrastructure, morphology and crystal growth of biogenic and synthetic apatites. , 1990, Connective tissue research.

[26]  O. B. Pedersen.  Thermoelasticity and plasticity of composites—I. Mean field theory , 1983 .

[27]  G. Pharr,et al.  Elastic properties of human cortical and trabecular lamellar bone measured by nanoindentation. , 1997, Biomaterials.

[28]  E. Eanes,et al.  The Effect of Fluoride on the Size and Morphology of Apatite Crystals Grown from Physiologic Solutions , 1998, Calcified Tissue International.

[29]  N. Guzelsu,et al.  A shear-lag model to account for interaction effects between inclusions in composites reinforced with rectangular platelets , 2000 .

[30]  P Zioupos,et al.  The role of collagen in the declining mechanical properties of aging human cortical bone. , 1999, Journal of biomedical materials research.

[31]  A. Boskey.  Noncollagenous matrix proteins and their role in mineralization. , 1989, Bone and mineral.

[32]  C. Rey,et al.  The carbonate environment in bone mineral: A resolution-enhanced fourier transform infrared spectroscopy study , 1989, Calcified Tissue International.

[33]  M. Glimcher,et al.  Shape and size of isolated bone mineralites measured using atomic force microscopy , 2001, Journal of orthopaedic research : official publication of the Orthopaedic Research Society.

[34]  S Lees,et al.  Considerations regarding the structure of the mammalian mineralized osteoid from viewpoint of the generalized packing model. , 1987, Connective tissue research.

[35]  G. H. Nancollas,et al.  The seeded growth of calcium phosphates. The kinetics of growth of dicalcium phosphate dihydrate on hydroxyapatite , 1976, Calcified Tissue Research.

[36]  P. Fratzl,et al.  Validation of quantitative backscattered electron imaging for the measurement of mineral density distribution in human bone biopsies. , 1998, Bone.

[37]  K. Mclauchlan,et al.  The hydration structure of collagen , 1969, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[38]  A J Bailey,et al.  Collagen cross-links in mineralizing tissues: a review of their chemistry, function, and clinical relevance. , 1998, Bone.

[39]  Handschin Rg,et al.  Crystallographic and chemical analysis of human bone apatite (Crista Iliaca). , 1994 .

[40]  Klaus Klaushofer,et al.  Nucleation and growth of mineral crystals in bone studied by small-angle X-ray scattering , 1991, Calcified Tissue International.

[41]  A. Burstein,et al.  The elastic and ultimate properties of compact bone tissue. , 1975, Journal of biomechanics.

[42]  J. J. Freeman,et al.  Raman Spectroscopic Detection of Changes in Bioapatite in Mouse Femora as a Function of Age and In Vitro Fluoride Treatment , 2001, Calcified Tissue International.

[43]  L. Quarles,et al.  Pathophysiology of X-linked hypophosphatemia, tumor-induced osteomalacia, and autosomal dominant hypophosphatemia: a perPHEXing problem. , 2001, The Journal of clinical endocrinology and metabolism.

[44]  R. D. Ray,et al.  Bone Density and Composition , 1966 .

[45]  P. Fratzl,et al.  Collagen packing and mineralization. An x-ray scattering investigation of turkey leg tendon. , 1993, Biophysical journal.

[46]  J. Katz,et al.  Elastic properties of apatites , 1982 .

[47]  J. Mammone,et al.  Micromechanics of bone strength and fracture. , 1993, Journal of biomechanics.

[48]  W C Van Buskirk,et al.  A continuous wave technique for the measurement of the elastic properties of cortical bone. , 1984, Journal of biomechanics.

[49]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[50]  R. Martin,et al.  The effects of collagen fiber orientation, porosity, density, and mineralization on bovine cortical bone bending properties. , 1993, Journal of biomechanics.

[51]  M. Pineri,et al.  Water–collagen interactions: Calorimetric and mechanical experiments , 1978, Biopolymers.

[52]  G. Pharr,et al.  Effects of anisotropy on elastic moduli measured by nanoindentation in human tibial cortical bone. , 2001, Journal of biomedical materials research.