Image Analysis of Curvature Using Classical Mechanics, the Elastica

  • Charles WilsonEmail author
  • James Dawson
Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)


The primary loading condition for implanted leads and other slender/compliant implantable medical devices is bending. Curvature, which is the inverse of radius of curvature, quantifies the magnitude of bending deformation and is used to motivate bench tests as well as used for computational modeling. The Association for the Advancement of Medical Instrumentation (AAMI) Transvenous Leads Working Group is currently studying how to best measure the curvature of leads under deformation, both in-vivo and on the bench.

Deformed lead segments bent to varying levels of severity, were imaged and the resultant curvature was analyzed using several optical analysis methods, including fitting circles, ellipses and splines to the deformed shape. Of all the methods studied, the one which showed the least amount of variability while also not over parameterizing was using the Elastica.

The Elastica, which was first formulated by James Bernoulli in 1691 and later solved by Leonhard Euler in 1744, is the closed form solution to large scale deformations of buckled structures. It is an example of bifurcation theory in the field of solid mechanics. The Elastica can take many different shapes while only having a few degrees of freedom. More importantly, the Elastica is the exact analytical solution to a long, slender beam under large deformation—which is precisely what is being studied. This approach has been implemented in an R script that allows the user to load an image, identify points in the image to which the Elastica should be fit, and optimizes the fit parameters using a least squares regression. Outputs from the script are the magnitude and location of the maximum curvature, the moment arm distance, and a goodness of fit metric.


Image analysis Bending Buckling Curvature Elastica 



The authors of this publication would like to acknowledge these individuals for their contribution to this study.

J. Splett, D. McColskey, T. Luther, D. Smith, N. Duraiswamy, and S. Raymond for conducting the curvature measurements and A. Himes for his technical consult.


  1. 1.
    Levien, R.: The elastica: a mathematical history. EECS Department, University of California, Berkeley, Technical Report No. UCB/EECS-2008-103 (2008)Google Scholar

Copyright information

© The Society for Experimental Mechanics, Inc. 2019

Authors and Affiliations

  1. 1.Medtronic, Inc.MinneapolisUSA

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