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Research Papers

Monte Carlo Type Simulations of Mineralized Collagen Fibril Based on Two Scale Asymptotic Homogenization

[+] Author and Article Information
Abhilash Awasthi

School of Engineering,
Indian Institute of Technology Mandi,
Kamand,
Mandi, Himachal Pradesh 175005, India
e-mail: abhiawasthi1993@gmail.com

Rajneesh Sharma

School of Engineering,
Indian Institute of Technology Mandi,
Kamand,
Mandi, Himachal Pradesh 175005, India
e-mail: rajnish.iitmandi@gmail.com

Rajesh Ghosh

School of Engineering,
Indian Institute of Technology Mandi,
Kamand,
Mandi, Himachal Pradesh 175005, India
e-mail: rajesh@iitmandi.ac.in

1Corresponding author.

Manuscript received June 4, 2018; final manuscript received December 30, 2018; published online February 13, 2019. Assoc. Editor: Jeffrey Ruberti.

J Biomech Eng 141(4), 041002 (Feb 13, 2019) (11 pages) Paper No: BIO-18-1259; doi: 10.1115/1.4042439 History: Received June 04, 2018; Revised December 30, 2018

A multiscale model for mineralized collagen fibril (MCF) is proposed by taking into account the uncertainties associated with the geometrical properties of the mineral phase and its distribution in the organic matrix. The asymptotic homogenization approach along with periodic boundary conditions has been used to derive the effective elastic moduli of bone's nanostructure at two hierarchical length scales, namely: microfibril (MF) and MCF. The uncertainties associated with the mineral plates have been directly included in the finite element mesh by randomly varying their sizes and structural arrangements. A total of 100 realizations for the MCF model with random distribution have been generated using an in-house MATLAB code, and Monte Carlo type of simulations have been performed under tension load to obtain the statistical equivalent modulus. The deformation response has been studied in both small (10%) and large (10%) strain regimes. The stress transformation mechanism has also been explored in MF which showed stress relaxation in the organic phase upon different stages of mineralization. The elastic moduli for MF under small and large strains have been obtained as 1.88 and 6.102 GPa, respectively, and have been used as an input for the upper scale homogenization procedure. Finally, the characteristic longitudinal moduli of the MCF in the small and large strain regimes are obtained as 4.08 ± 0.062 and 12.93 ± 0.148 GPa, respectively. All the results are in good agreement to those obtained from previous experiments and molecular dynamics (MD) simulations in the literature with a significant reduction in the computational cost.

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Figures

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Fig. 1

Two scales hierarchical structure of the MCF: (a) MCF scale with mineral plates embedded in the MF matrix and (b) MF scale as TC molecules embedded in a matrix of NCPs and water

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Fig. 2

General FE model with periodic boundary conditions

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Fig. 3

Two-dimensional MF models: (a) shows the unmineralized MF model (MF1), (b) shows the MF model with mineral plates in the axial gap between two TC molecules (MF2), and (c) shows the mineral plates being deposited in the gap and extra collagenous space (MF3)

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Fig. 4

Different 2D MCF models. The figure shows models with (a) regular distribution (F1), (b) staggered distribution (F2), and (c) random distribution (F3) of mineral plates within the MF matrix.

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Fig. 5

Two-dimensional MF models with their respective longitudinal stress S11 (MPa) contour plots in the small strain regime. (a) shows the unmineralized MF model (MF1), (b) shows the MF model with mineral plates in the axial gap between two TC molecules (MF2), and (c) shows the mineral plates being deposited in the gap and extra collagenous space (MF3).

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Fig. 6

Young's modulus at SS and LS for different MF models from the present study and their comparison with the values from the literature. (a) shows the small and large strain moduli for different MF models and (b) compares the elastic modulus from the present study with the values from the literature.

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Fig. 7

Different 2D MCF models with their respective longitudinal stress S11 (MPa) contour plots. The figure shows MCF models with (a) regular distribution (F1), (b) staggered distribution (F2), and (c) random distribution (F3) of mineral plates within the MF matrix.

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Fig. 8

Effect of the sample number on the mean elastic modulus of MCF under large strain

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Fig. 9

Young's modulus at small strain (ESS) and large strain (ELS) for different MCF models and their comparison with the literature. (a) shows the results of the present study for small and large strain moduli, and (b) compares the small strain elastic modulus values with the literature. (c) and (d) shows the small and large strain elastic modulus histograms, respectively, for model F3.

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Fig. 10

Weibull plot for the elastic modulus data from MCF random model F3 for small strain (SS) and large strain (LS) regimes

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Fig. 11

Effects on the upper scale (MCF) characteristic modulus due to the first and second stage mineralization at the lower (MF) scale

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Fig. 12

Parametric study on the MCF random model: (a) shows the histogram for models having variation in length of mineral plates, (b) shows the histogram for models having variation in the axial gap between mineral plates, and (c) shows the histogram for models having variation in lateral spacing between the mineral plates

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