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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

Abstract

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.

Keywords

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

Notes

Acknowledgments

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)

References

  1. 1.
    Hall MS et al (2013) Toward single cell traction microscopy within 3D collagen matrices. Exp Cell Res 319(16):2396–2408CrossRefGoogle Scholar
  2. 2.
    DuFort CC, Paszek MJ, Weaver VM (2011) Balancing forces: architectural control of mechanotransduction. Nat Rev Mol Cell Biol 12(5):308–319CrossRefGoogle Scholar
  3. 3.
    Paszek MJ et al (2005) Tensional homeostasis and the malignant phenotype. Cancer Cell 8(3):241–254MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kostic A, Lynch CD, Sheetz MP (2009) Differential matrix rigidity response in breast cancer cell lines correlates with the tissue tropism. PLoS One 4(7):e6361CrossRefGoogle Scholar
  5. 5.
    Culver JC, Dickinson ME (2010) The effects of hemodynamic force on embryonic development. Microcirculation 17(3):164–178CrossRefGoogle Scholar
  6. 6.
    Tan JL et al (2003) Cells lying on a bed of microneedles: an approach to isolate mechanical force. Proc Natl Acad Sci 100(4):1484–1489CrossRefGoogle Scholar
  7. 7.
    Dembo M, Wang Y-L (1999) Stresses at the cell-to-substrate interface during locomotion of fibroblasts. Biophys J 76(4):2307–2316CrossRefGoogle Scholar
  8. 8.
    Legant WR et al (2010) Measurement of mechanical tractions exerted by cells in three-dimensional matrices. Nat Methods 7(12):969–971CrossRefGoogle Scholar
  9. 9.
    Griffith LG, Swartz MA (2006) Capturing complex 3D tissue physiology in vitro. Nat Rev Mol Cell Biol 7(3):211–224CrossRefGoogle Scholar
  10. 10.
    Cukierman E et al (2001) Taking cell-matrix adhesions to the third dimension. Science 294(5547):1708–1712CrossRefGoogle Scholar
  11. 11.
    Maniatty A, Zabaras N, Stelson K (1989) Finite element analysis of some inverse elasticity problems. J Eng Mech 115(6):1303–1317CrossRefGoogle Scholar
  12. 12.
    Zabaras N, Morellas V, Schnur D (1989) Spatially regularized solution of inverse elasticity problems using the BEM. Commun Appl Numer Methods 5(8):547–553CrossRefzbMATHGoogle Scholar
  13. 13.
    Gjorevski N, Nelson CM (2012) Mapping of mechanical strains and stresses around quiescent engineered three-dimensional epithelial tissues. Biophys J 103(1):152–162CrossRefGoogle Scholar
  14. 14.
    Koch TM et al (2012) 3D traction forces in cancer cell invasion. PLoS One 7(3):e33476CrossRefGoogle Scholar
  15. 15.
    Butler JP et al (2002) Traction fields, moments, and strain energy that cells exert on their surroundings. Am J Physiol-Cell Physiol 282(3):C595–C605CrossRefGoogle Scholar
  16. 16.
    Storm C et al (2005) Nonlinear elasticity in biological gels. Nature 435(7039):191–194CrossRefGoogle Scholar
  17. 17.
    Franck C et al (2007) Three-dimensional full-field measurements of large deformations in soft materials using confocal microscopy and digital volume correlation. Exp Mech 47(3):427–438CrossRefGoogle Scholar
  18. 18.
    Hadamard J (1902) Sur les problèmes aux dérivées partielles et leur signification physique. Princet Univ Bull 13(49–52):28Google Scholar
  19. 19.
    Hofmann B, Scherzer O (1998) Local ill-posedness and source conditions of operator equations in Hilbert spaces. Inverse Probl 14(5):1189MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Eremeyev VA, Lebedev LP (2013) Existence of weak solutions in elasticity. Math Mech Solids 18(2):204–217MathSciNetCrossRefGoogle Scholar
  21. 21.
    Gurtin ME (1973) The linear theory of elasticity, in linear theories of elasticity and thermoelasticity. Springer, Berlin p 1-295Google Scholar
  22. 22.
    Desoer C, Whalen B (1963) A note on pseudoinverses. J Soc Ind Appl Math 11(2):442–447MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Frommer A, Maass P (1999) Fast CG-based methods for Tikhonov–Phillips regularization. SIAM J Sci Comput 20(5):1831–1850MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ben-Israel A (2003) Generalized inverses : theory and applications. In: Greville TNE (ed), Springer-Verlag, New YorkGoogle Scholar
  25. 25.
    Nashed MZ (1987) A new approach to classification and regularization of ill-posed operator equations. In: Engl HW, Groetsch CW (eds) Inverse and Ill-posed problems, notes and reports in mathematics in science and engineering, vol 4. Academic, New York, pp 53–75Google Scholar
  26. 26.
    Cheney E, Kincaid, D (2007) Numerical Mathematics and Computing. Cengage LearningGoogle Scholar
  27. 27.
    Gould NI, Hribar ME, Nocedal J (2001) On the solution of equality constrained quadratic programming problems arising in optimization. SIAM J Sci Comput 23(4):1376–1395MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Calvetti D et al (2000) Tikhonov regularization and the L-curve for large discrete ill-posed problems. J Comput Appl Math 123(1):423–446MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hansen PC (1992) Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev 34(4):561–580MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hansen PC, O’Leary DP (1993) The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J Sci Comput 14(6):1487–1503MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Hestenes MR, Stiefel E (1952) Methods of conjugate gradients for solving linear systems. NBS 49:409–436MathSciNetzbMATHGoogle Scholar
  32. 32.
    Bangerth W et al (2015) The deal.II Library, version 8.2. Arch Numer Softw 3:1–8. doi: 10.11588/ans.2015.100.18031
  33. 33.
    Tomei AA et al (2009) 3D collagen cultures under well-defined dynamic strain: a novel strain device with a porous elastomeric support. Biotechnol Bioeng 103(1):217–225MathSciNetCrossRefGoogle Scholar
  34. 34.
    Barocas VH, Moon AG, Tranquillo RT (1995) The fibroblast-populated collagen microsphere assay of cell traction force—part 2: measurement of the cell traction parameter. J Biomech Eng 117(2):161–170CrossRefGoogle Scholar
  35. 35.
    Wakitani S et al (1998) Repair of large full-thickness articular cartilage defects with allograft articular chondrocytes embedded in a collagen gel. Tissue Eng 4(4):429–444CrossRefGoogle Scholar
  36. 36.
    Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. In: Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences. The Royal SocietyGoogle Scholar
  37. 37.
    Mura T (2012) Micromechanics of defects in solids, vol 3. Springer Science & Business Media, BerlinGoogle Scholar
  38. 38.
    Trickey WR et al (2006) Determination of the Poisson’s ratio of the cell: recovery properties of chondrocytes after release from complete micropipette aspiration. J Biomech 39(1):78–87MathSciNetCrossRefGoogle Scholar
  39. 39.
    Lindström SB et al (2013) Finite-strain, finite-size mechanics of rigidly cross-linked biopolymer networks. Soft Matter 9(30):7302–7313CrossRefGoogle Scholar
  40. 40.
    Staunton JR (2014) Mechanics of cancer cells in 3D microenvironments. Arizona State University, TempeGoogle Scholar
  41. 41.
    Meng C, Heltsley W, Pollard DD (2012) Evaluation of the Eshelby solution for the ellipsoidal inclusion and heterogeneity. Comput Geosci 40:40–48Google Scholar
  42. 42.
    Geuzaine C, Remacle JF (2009) Gmsh: a 3-D finite element mesh generator with built-in pre-and post-processing facilities. Int J Numer Meth Eng 79(11):1309–1331MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Vader D et al (2009) Strain-induced alignment in collagen gels. PloS one 4(6):e5902CrossRefGoogle Scholar
  44. 44.
    Janmey PA et al (2007) Negative normal stress in semiflexible biopolymer gels. Nat Mater 6(1):48–51CrossRefGoogle Scholar
  45. 45.
    Mofrad MR (2009) Rheology of the cytoskeleton. Ann Rev Fluid Mech 41:433–453CrossRefzbMATHGoogle Scholar
  46. 46.
    Karcher H et al (2003) A three-dimensional viscoelastic model for cell deformation with experimental verification. Biophys J 85(5):3336–3349CrossRefGoogle Scholar
  47. 47.
    Yoshigi M et al (2005) Mechanical force mobilizes zyxin from focal adhesions to actin filaments and regulates cytoskeletal reinforcement. J Cell Biol 171(2):209–215CrossRefGoogle Scholar
  48. 48.
    De Borst R et al (2012) Nonlinear finite element analysis of solids and structures. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  49. 49.
    Bertsekas DP (1999) Nonlinear programming. Athena Scientific, BelmontzbMATHGoogle Scholar
  50. 50.
    Jacoby SL, Kowalik JS, Pizzo JT (1972) Iterative methods for nonlinear optimization problems. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  51. 51.
    Hager WW, Zhang H (2006) A survey of nonlinear conjugate gradient methods. Pac J Optim 2(1):35–58MathSciNetzbMATHGoogle Scholar
  52. 52.
    Boas RP, Boas HP (1996) A primer of real functions, vol 13. Cambridge University Press, CambridgezbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringCornell UniversityIthacaUSA

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