Implementation of a Modified Moving Least Squares Approximation for Predicting Soft Tissue Deformation Using a Meshless Method

  • Habibullah Amin Chowdhury
  • Grand Roman Joldes
  • Adam Wittek
  • Barry Doyle
  • Elena Pasternak
  • Karol Miller

Abstract

In applications where the organic soft tissue undergoes large deformations, traditional finite element methods can fail due to element distortion. In this context, meshless methods, which require no mesh for defining the interpolation field, can offer stable solutions. In meshless method, the moving least square (MLS) shape functions have been widely used for approximating the unknown field functions using the scattered field nodes. However, the classical MLS places strict requirements on the nodal distributions inside the support domain in order to maintain the non-singularity of the moment matrix. These limitations are preventing the practical use of higher order polynomial basis in classical MLS for randomly distributed nodes despite their capability for more accurate approximation of complex deformation fields. A modified moving least squares (MMLS) approximation has been recently developed by ISML. This paper assesses the interpolation capabilities of the MMLS. The proposed meshless method based on MMLS is used for computing the extension of a soft tissue sample and for a brain deformation simulation in 2D. The results are compared with the commercial finite element software ABAQUS. The simulation results demonstrate the superior performance of the MMLS over classical MLS with linear basis functions in terms of accuracy of the solution.

References

  1. 1.
    Cotin, S., Delingette, H., Ayache, N.: A hybrid elastic model for real-time cutting, deformations, and force feedback for surgery training and simulation. Vis. Comput. 16(8), 437-452 (2000). doi:10.1007/Pl00007215
  2. 2.
    Warfield, S.K., Talos, F., Tei, A., Bharatha, A., Nabavi, A., Ferrant, M., Black, P.M.L., Jolesz, F.A., Kikinis, R.: Real-time registration of volumetric brain MRI by biomechanical simulation of deformation during image guided neurosurgery. Comput. Vis. Sci. 5(1), 3–11 (2002). doi:10.1007/s00791-002-0083-7 CrossRefMATHGoogle Scholar
  3. 3.
    Belytschko, T., Lu, Y.Y., Gu, L.: Element-free Galerkin methods. Int. J. Numer. Methods. Eng. 37(2), 229–256 (1994). doi:10.1002/nme.1620370205
  4. 4.
    Liu, G-R.: Meshfree Methods: Moving Beyond the Finite Element Method. CRC Press, Boca Raton, Florida, USA (2010)Google Scholar
  5. 5.
    Wittek, A., Joldes, G.R., Miller, K.: Algorithms for computational biomechanics of the brain. In Miller, K. (Ed), Biomechanics of the Brain, pp. 189-219, Springer, New York (2011)Google Scholar
  6. 6.
    Miller, K., Wittek, A., Joldes, G.R.: Biomechanical Modeling of the Brain for Computer-Assisted Neurosurgery. In Miller, K. (Ed), Biomechanics of the Brain, pp. 111-136, Springer, New York (2011)Google Scholar
  7. 7.
    Nayroles, B., Touzot, G., Villon, P.: Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech. 10(5), 307–318 (1992). doi:10.1007/bf00364252 CrossRefMATHGoogle Scholar
  8. 8.
    Shepard, D.: A two-dimensional interpolation for irregularly-spaced data. Paper presented at the 23rd ACM national conference (1968)Google Scholar
  9. 9.
    Nguyen, V.P., Rabczuk, T., Bordas, S., Duflot, M.: Meshless methods: a review and computer implementation aspects. Math. Comput. Simul. 79(3), 763–813 (2008). doi:10.1016/j.matcom.2008.01.003 CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Lancaster, P., Salkauskas, K.: Surfaces generated by moving least-squares methods. Math. Comput. 37(155), 141–158 (1981). doi:10.2307/2007507
  11. 11.
    Li, S., Liu, W.K.: Meshfree Particle Methods. Springer, Berlin (2004)Google Scholar
  12. 12.
    Joldes, G.R., Chowdhury, H.A., Wittek, A., Doyle, B., Miller, K.: Modified Moving Least Squares with Polynomial Bases for Scattered Data Approximation. UWA, Perth, WA, Report # ISML/02/2014: 17 pages. (school.mech.uwa.edu.au/ISML/index.php/Reports) (2014)Google Scholar
  13. 13.
    Fasshauer, G.E.: Meshfree Approximation Methods with MATLAB. World Scientific Publishing Co., Inc., Singapore (2007)Google Scholar
  14. 14.
    Miller, K., Joldes, G., Lance, D., Wittek, A.: Total Lagrangian explicit dynamics finite element algorithm for computing soft tissue deformation. Commun. Numer. Methods. Eng. 23(2), 121–134 (2006). doi:10.1002/cnm.887 CrossRefMathSciNetGoogle Scholar
  15. 15.
    Joldes, G.R., Wittek, A., Couton, M., Warfield, S.K., Miller, K.: Real-Time Prediction of Brain Shift Using Nonlinear Finite Element Algorithms. Medical Image Computing and Computer-Assisted Intervention – Miccai 2009, Pt Ii, Proceedings, vol. 5762, pp. 300–307 (2009)Google Scholar
  16. 16.
    Joldes, G.R., Wittek, A., Miller, K.: Computation of intra-operative brain shift using dynamic relaxation. Comput. Methods. Appl. Mech. Eng. 198(41–44), 3313–3320 (2009). doi:10.1016/j.cma.2009.06.012
  17. 17.
    Joldes, G.R., Wittek, A., Miller, K.: Suite of finite element algorithms for accurate computation of soft tissue deformation for surgical simulation. Med. Image. Anal. 13(6), 912–919 (2009). doi:10.1016/j.media.2008.12.001
  18. 18.
    Joldes, G.R., Wittek, A., Miller, K.: Real-time nonlinear finite element computations on GPU – application to neurosurgical simulation. Comput. Methods. Appl. Mech. Eng. 199(49–52), 3305–3314 (2010). doi:10.1016/j.cma.2010.06.037 CrossRefMATHGoogle Scholar
  19. 19.
    Wittek, A., Joldes, G., Couton, M., Warfield, S.K., Miller, K. Patient-specific non-linear finite element modelling for predicting soft organ deformation in real-time; application to non-rigid neuroimage registration. Prog. Biophys. Mol. Biol. 103(2–3), 292–303 (2010). doi:10.1016/j.pbiomolbio.2010.09.001
  20. 20.
    Horton, A., Wittek, A., Joldes, G.R., Miller, K.: A meshless total Lagrangian explicit dynamics algorithm for surgical simulation. Int. J. Numer. Methods Biomed. Eng. 26(8), 977–998 (2010). doi:10.1002/cnm.1374 CrossRefMATHGoogle Scholar
  21. 21.
    Most, T., Bucher, C.: A moving least squares weighting function for the element-free Galerkin method which almost fulfills essential boundary conditions. Struct. Eng. Mech. 21(3), 315–332 (2005)CrossRefGoogle Scholar
  22. 22.
    Joldes, G.R, Wittek, A., Miller, K.: An adaptive dynamic relaxation method for solving nonlinear finite element problems. Application to brain shift estimation. Int. J. Numer. Methods. Biomed. Eng. 27(2), 173–185 (2011). doi:10.1002/cnm.1407
  23. 23.
    Miller, K., Chinzei, K., Orssengo, G., Bednarz, P.: Mechanical properties of brain tissue in-vivo: experiment and computer simulation. J. Biomech. 33(11), 1369–1376 (2000). doi:10.1016/S0021-9290(00)00120-2
  24. 24.
    Zhang, J.Y., Joldes, G.R., Wittek, A., Miller, K.: Patient-specific computational biomechanics of the brain without segmentation and meshing. Int. J. Numer. Methods. Biomed. Eng. 29(2), 293–308 (2013). doi:10.1002/cnm.2507 CrossRefMathSciNetGoogle Scholar
  25. 25.
    Miller, K.: Biomechanics of the Brain. Springer, New York (2011)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Habibullah Amin Chowdhury
    • 1
  • Grand Roman Joldes
    • 1
  • Adam Wittek
    • 1
  • Barry Doyle
    • 1
  • Elena Pasternak
    • 2
  • Karol Miller
    • 1
  1. 1.Intelligent Systems for Medicine Laboratory (ISML)The University of Western AustraliaCrawleyAustralia
  2. 2.School of Mechanical and Chemical EngineeringThe University of Western AustraliaCrawleyAustralia

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