Computational Mechanics

, Volume 58, Issue 1, pp 91–105 | Cite as

Force sensing using 3D displacement measurements in linear elastic bodies

  • Xinzeng FengEmail author
  • Chung-Yuen Hui
Original Paper


In cell traction microscopy, the mechanical forces exerted by a cell on its environment is usually determined from experimentally measured displacement by solving an inverse problem in elasticity. In this paper, an innovative numerical method is proposed which finds the “optimal” traction to the inverse problem. When sufficient regularization is applied, we demonstrate that the proposed method significantly improves the widely used approach using Green’s functions. Motivated by real cell experiments, the equilibrium condition of a slowly migrating cell is imposed as a set of equality constraints on the unknown traction. Our validation benchmarks demonstrate that the numeric solution to the constrained inverse problem well recovers the actual traction when the optimal regularization parameter is used. The proposed method can thus be applied to study general force sensing problems, which utilize displacement measurements to sense inaccessible forces in linear elastic bodies with a priori constraints.


Inverse problem Finite element Linear elasticity  Cell traction microscopy Green’s function Conjugate gradient method 



XZF is partially supported by the National Center for Research Resources (5R21RR025801-03). Both authors give thanks to Dr. Mingming Wu and Matthew S. Hall for motivation of the inverse problem in cell traction microscopy. XZF gives thanks to Dr. Timothy J. Healey and Chenxi Wu for helpful discussions on the proof of the convergence theorem in the appendix.

Supplementary material

466_2016_1283_MOESM1_ESM.docx (1.6 mb)
Supplementary material 1 (docx 1592 KB)


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringCornell UniversityIthacaUSA

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