Technical Briefs

Hermitian Splines for Modeling Biological Soft Tissue Systems That Exhibit Nonlinear Force-Elongation Curves

[+] Author and Article Information
F. Martel, M. Denninger, E. Langelier, M-C. Turcotte

 Groupe de recherche en performance et sécurité humaine de l′Université de Sherbrooke, Université de Sherbrooke,2500 Boul. de l’Université, Sherbrooke, QC, Canada, J1K 2R1

D. Rancourt1

 Groupe de recherche en performance et sécurité humaine de l′Université de Sherbrooke, Université de Sherbrooke,2500 Boul. de l’Université, Sherbrooke, QC, Canada, J1K 2R1Denis.Rancourt@usherbrooke.ca


Corresponding author.

J Biomech Eng 133(9), 094505 (Oct 11, 2011) (5 pages) doi:10.1115/1.4004949 History: Received July 20, 2011; Accepted August 26, 2011; Published October 11, 2011; Online October 11, 2011

Numerical simulation of soft tissue mechanical properties is a critical step in developing valuable biomechanical models of live organisms. A cubic Hermitian spline optimization routine is proposed in this paper to model nonlinear experimental force-elongation curves of soft tissues, in particular when modeled as lumped elements. Boundary conditions are introduced to account for the positive definiteness and the particular curvature of the experimental curve to be fitted. The constrained least-square routine minimizes user intervention and optimizes fitting of the experimental data across the whole fitting range. The routine provides coefficients of a Hermitian spline or corresponding knots that are compatible with a number of constraints that are suitable for modeling soft tissue tensile curves. These coefficients or knots may become inputs to user-defined component properties of various modeling software. Splines are particularly advantageous over the well-known exponential model to account for the traction curve flatness at low elongations and to allow for more flexibility in the fitting process. This is desirable as soft tissue models begin to include more complex physical phenomena.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 1

Problems involved in fitting cubic splines to a soft tissue force/elongation or stress-strain curve

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Figure 2

Definition of spline segments

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Figure 3

Experimental force-elongation curve of a rat tail tendon obtained from a traction test. Curve is fitted with three different spline methods: one resulting from our own routine, one from using our routine output knots into the MSC.Adams routine, and one from using the MSC.Adams routine with user-selected experimental data points (one every 75 points).

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Figure 4

Fitting errors (as defined per Eq. 9) for various positions of the inflection point and the number of spline segments

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Figure 5

Typical force-deformation curve of a foam pad under planar compression exhibiting an S-shape. Deformation is in percent of original foam pad thickness.



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