Research Papers

Directional Tortuosity as a Predictor of Modulus Damage for Vertebral Cancellous Bone

[+] Author and Article Information
David P. Fyhrie

Department of Orthopaedic Surgery,
University of California-Davis Medical Center,
4635 2nd Avenue, Suite 2000,
Sacramento, CA 95817
Department of Biomedical Engineering,
University of California Davis,
4635 2nd Avenue, Suite 2000,
Sacramento, CA 95817
e-mail: dpfyhrie@ucdavis.edu

Roger Zauel

Bone and Joint Center,
Henry Ford Health System,
Detroit, MI 48202

1Corresponding author.

Manuscript received June 30, 2014; final manuscript received November 17, 2014; accepted manuscript posted November 20, 2014; published online December 10, 2014. Assoc. Editor: Ara Nazarian.

J Biomech Eng 137(1), 011007 (Jan 01, 2015) (6 pages) Paper No: BIO-14-1304; doi: 10.1115/1.4029177 History: Received June 30, 2014; Revised November 17, 2014; Accepted November 20, 2014; Online December 10, 2014

There are many methods used to estimate the undamaged effective (apparent) moduli of cancellous bone as a function of bone volume fraction (BV/TV), mean intercept length (MIL), and other image based average microstructural measures. The MIL and BV/TV are both only functions of the cancellous microstructure and, therefore, cannot directly account for damage induced changes in the intrinsic trabecular hard tissue mechanical properties. Using a nonlinear finite element (FE) approximation for the degradation of effective modulus as a function of applied effective compressive strain, we demonstrate that a measurement of the directional tortuosity of undamaged trabecular hard tissue strongly predicts directional effective modulus (r2> 0.90) and directional effective modulus degradation (r2> 0.65). This novel measure of cancellous bone directional tortuosity has the potential for development into an anisotropic approach for calculating effective mechanical properties as a function of trabecular level material damage applicable to understanding how tissue microstructure and intrinsic hard tissue moduli interact to determine cancellous bone quality.

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Fig. 1

A 5 mm cubical FE model with 50 μm voxels to illustrate apparent and intrinsic properties

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Fig. 2

Connections between points on two parallel lines (“loading platens”) illustrating how different connections are ideal (T = 1) or less than ideal (T > 1). Tortuosity (T) is defined as the line length between the points divided by the distance between the loading platens.

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Fig. 3

An illustration using a 2D “trabecular bone specimen” of how to calculate average tortuosity. For all point pairs A–B on the loading planes (platens): average the ratio of the shortest distance between points to the distance between the planes.

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Fig. 4

2D illustration for the distance matrix. For node “n3,” the distance matrix is the list of nodes and length of the connection. For example, “n3, n7, 21/2d” is the connection from the center to the upper right. The list of all connections for all 3 × 3 regions centered on a filled pixel is the distance matrix of a 2D binary image.

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Fig. 5

Model stiffness (for all models) was strongly separated by load direction, but was closely related to tortuosity. The contour lines are the density of points for models of the XX (blue; lower right) and ZZ (red; upper left) directions. Lines are splines to show the general trend.

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Fig. 6

Stiffness (damaged and undamaged) was a strong power function of tortuosity independent of direction. Dark markers are undamaged and light are damaged. Damage moved the damaged points roughly parallel to the regression line.

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Fig. 7

The change in stiffness was strongly predicted by change in tortuosity for both directions. Solid markers are for the X direction and open circles are for the Z direction. The exponential fit suggests that the increase in tortuosity has a nonlinear effect on stiffness.




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