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TECHNICAL PAPERS: Cell

Elastic Model for Crimped Collagen Fibrils

[+] Author and Article Information
Alan D. Freed

Bio Sciences and Technology Branch,  NASA’s John H. Glenn Research Center at Lewis Field, 21000 Brookpark Road, Cleveland, OH 44135 and Adjunct Staff, Department of Biomedical Engineering,  The Cleveland Clinic Foundation, 9500 Euclid, Avenue, Cleveland, OH 44195alan.d.freed@nasa.gov

Todd C. Doehring

Department of Biomedical Engineering, ND-20, Lerner Research Institute,  The Cleveland Clinic Foundation, 9500 Euclid Avenue, Cleveland, OH 44195doehrint@ccf.org

J Biomech Eng 127(4), 587-593 (Feb 01, 2005) (7 pages) doi:10.1115/1.1934145 History: Received May 18, 2004; Revised February 01, 2005

A physiologic constitutive expression is presented in algorithmic format for the nonlinear elastic response of wavy collagen fibrils found in soft connective tissues. The model is based on the observation that crimped fibrils in a fascicle have a three-dimensional structure at the micron scale that we approximate as a helical spring. The symmetry of this wave form allows the force/displacement relationship derived from Castigliano’s theorem to be solved in closed form: all integrals become analytic. Model predictions are in good agreement with experimental observations for mitral-valve chordæ tendineæ.

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Copyright © 2005 by American Society of Mechanical Engineers
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Figures

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Figure 1

Scanning electron microscopy (SEM) photograph of a cross-sectional cut of a chordæ tendineæ taken from a porcine mitral valve showing a three-dimensional (3D) undulating fiber structure. (Reproduced with permission from J. Liao (6).)

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Figure 2

A schematic of the stress/stretch response of collagenous tissues and how their wavy structures vary with deformation

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Figure 3

SEM photograph showing the helical nature of crimped collagen fibrils in chordæ tendineæ taken from a porcine mitral valve. (Reproduced with permission from J. Liao (6).)

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Figure 4

Diagram of how force P acting along the centerline of a helix is transferred to forces F and V and moments M and T that act along the backbone of the helix

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Figure 5

Plots of crimp wavelength H0 normalized by fibril radius r0 (where in one curve H0 varies, and in the other curve r0 varies) vs the elastic modulus E normalized by the secant modulus E¯ evaluated at the transition between heel and linear regions

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Figure 6

Plots of the amplitude R and angle of pitch ϕ of crimp (normalized by their initial values R0 and ϕ0) vs stretch λ over the toe and heel regions

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Figure 7

Dependence of Algorithm 1 on parameter R0∕r0, where λ¯ and σ¯=E¯(λ¯−1) are values for stretch and stress at the transition point between linear and nonlinear behaviors. The linear region, where λ>λ¯, is not shown.

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Figure 8

Dependence of Algorithm 1 on parameter H0∕r0, where λ¯ and σ¯=E¯(λ¯−1) are values for stretch and stress at the transition point between linear and nonlinear behaviors. The linear region, where λ>λ¯, is not shown.

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Figure 9

A fit of the helix model to data taken from five, porcine, mitral-value, chordæ-tendineæ specimens

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