Technical Briefs

A Continuous Method to Compute Model Parameters for Soft Biological Materials

[+] Author and Article Information
Martin L. Tanaka, Charles A. Weisenbach, Mark Carl Miller

 Department of Engineering and Technology, Western Carolina University, Cullowhee, NC 28723Department of Biology,  Clarkson University, Potsdam, NY 13699 Orthopaedic Biomechanics Laboratory, Allegheny General Hospital, Pittsburgh, PA 15212

Laurel Kuxhaus1

Department of Mechanical & Aeronautical Engineering,  Clarkson University, Potsdam, NY 13699 e-mail: lkuxhaus@clarkson.edu


Corresponding author.

J Biomech Eng 133(7), 074502 (Jul 22, 2011) (7 pages) doi:10.1115/1.4004412 History: Received March 07, 2011; Revised May 25, 2011; Posted June 13, 2011; Published July 22, 2011; Online July 22, 2011

Developing appropriate mathematical models for biological soft tissues such as ligaments, tendons, and menisci is challenging. Stress-strain behavior of these tissues is known to be continuous and characterized by an exponential toe region followed by a linear elastic region. The conventional curve-fitting technique applies a linear curve to the elastic region followed by a separate exponential curve to the toe region. However, this technique does not enforce continuity at the transition between the two regions leading to inaccuracies in the material model. In this work, a Continuous Method is developed to fit both the exponential and linear regions simultaneously, which ensures continuity between regions. Using both methods, three cases were evaluated: idealized data generated mathematically, noisy idealized data produced by adding random noise to the idealized data, and measured data obtained experimentally. In all three cases, the Continuous Method performed superiorly to the conventional technique, producing smaller errors between the model and data and also eliminating discontinuities at the transition between regions. Improved material models may lead to better predictions of nonlinear biological tissues’ behavior resulting in improved the accuracy for a large array of models and computational analyses used to predict clinical outcomes.

Copyright © 2011 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

Ideal data set fitted with both the Piecewise (a) and Continuous (b) Methods. The Piecewise Method (c) has a discontinuity in slope, dσ/dɛ. In contrast, the Continuous Method (d) has no observable discontinuity in the slope. Note that the discontinuity occurring when the Piecewise Method is applied occurs even with this ideal data set.

Grahic Jump Location
Figure 2

Noisy ideal data set with moderate magnitude noise (0.04). Like the ideal data set, a discontinuity is again observed for the Piecewise Method (a) and (c). Even in the presence of noise, the Continuous Method has both C0 and C1 continuity (b) and (d).

Grahic Jump Location
Figure 3

Ideal data set with high noise level (magnitude 0.4). At a noise level ten times greater than in Fig. 2, the Piecewise Method is unable to accurately model data points (a) and (d). However, the Continuous Method is still able to fit the data points moderately well.

Grahic Jump Location
Figure 4

Measured data obtained from tensile testing of the lateral porcine meniscus analyzed using the Piecewise Method (a) and the Continuous Method (b). Like the generated data sets previously analyzed, a C0 and C1 discontinuity occurred when the Piecewise Method was applied. The Continuous Method did not have a discontinuity (b) and (d). An additional observation was that the modeled data points (circles) closely approximated the measured data points (dots) even near the transition point.




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