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TECHNICAL PAPERS: Cell

# Elastic Deformation of Mineralized Collagen Fibrils: An Equivalent Inclusion Based Composite Model

[+] Author and Article Information
Ozan Akkus

ozan.akkus@utoledo.eduPh.DDepartment of Bioengineering,  The University of Toledo, 2801 W. Bancroft St., Mail Stop 303, Toledo, OH 43606-3390

J Biomech Eng 127(3), 383-390 (Jan 31, 2005) (8 pages) doi:10.1115/1.1894204 History: Received August 19, 2004; Revised November 29, 2004; Accepted January 31, 2005

## Abstract

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 $ν12(=ν21)$. 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.

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## Figures

Figure 1

The pattern of mineralization of collagen fibril assumed by the model (adopted from Fratzl (6)). The longer axis of the fibril is along the x3 axis and the plane transverse to the longer axis is defined by x1−x2 plane. Mineral crystals are denoted as black inclusions embedded in a collagen matrix shown in gray. Crystals have prolate ellipsoid shape and they appear as circles in the transverse plane and as ellipses in the longitudinal plane.

Figure 2

Changes in the technical constants of collagen fibers with increasing volume fraction of mineral crystals

Figure 3

Percent change in the technical constants of collagen fibers with increasing aspect ratio of mineral crystals at a mineral volume fraction of 0.42. The percent change is calculated with respect to the technical constant at the aspect ratio of 14 (vertical dashed line) which is calculated assuming that a mineral crystallite occupies the gap region between two successive collagen molecules.

Figure 4

The effect of mineral volume fraction on the partition of applied longitudinal (σ3=100MPa) or transverse (σ1=100MPa) normal stresses between mineral and collagen phases

Figure 5

The effect of mineral volume fraction on the partition of applied shear stress (100 MPa) between the mineral and collagen phases

Figure 6

The comparison of the predicted elastic modulus calculated along the longer axis of the fibril and those experimentally measured via nanoindentation (solid circles, 22.5 GPa-individual trabecula (38), 24.6 GPa osteons (36), 25.8 GPa-interstitial matrix (37), 28.2 GPa-interstitial matrix (36), all human bone) and macromechanical tests (circles, 17 GPa femur (49), 18.4 GPa femur (50), 20 GPa femur (51), human bone)

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