A Generative Model of Hyperelastic Strain Energy Density Functions for Real-Time Simulation of Brain Tissue Deformation

  • Alejandro GranadosEmail author
  • Martin Schweiger
  • Vejay Vakharia
  • Andrew W. McEvoy
  • Anna Miserocchi
  • John S. Duncan
  • Rachel Sparks
  • Sébastien Ourselin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11768)


Purpose.  The goal of this paper is to build a simulation environment that allows for the prediction of patient-specific tissue response by drawing samples from a generative model with a probability distribution. We propose a Gaussian Process (GP) regression approach to learn distributions over strain energy density functions including elastography, linear and hyperelastic models reported in the literature. Methods.  We gather a total of 73 models characterising elastic properties of brain white matter, grey matter and abnormalities and express them as strain energy density functions. A multi-output GP is used to quantify means and confidence intervals across each anatomical region and model. We sample the GP distribution and use nonlinear optimisation to fit a Neo-Hookean meta-model to guarantee stable strain energy functions. We validate the Neo-Hookean meta-model by fitting known strain energy density functions from the literature and report optimisation cost. We also validate the ability of the GP to approximate elastic properties of tissue given a reference deformed state using simulation. Results.  The GP was able to capture confidence intervals of varying strain ranges; the GP parameters and optimisation costs indicated a higher variability of hyperelastic models compared to elastography and linear models. Although one term is insufficient to fully capture hyperelastic models with higher number of terms, the resulting meta model is stable for real-time simulation within a wider range of stretches captured during mechanical characterisation of soft tissue. We demonstrated that our approach was able to approximate known elastic properties of tissue with a root-mean-squared error of 0.6 mm of node displacements when drawing six samples from a distribution of hyperelastic white matter. Conclusion.  In this initial proof-of-concept, we demonstrated a GP-based approach to estimate the elastic behaviour of brain tissue through simulation by sampling a generative model comprising elastic models found in the literature.


Biomechanics Brain deformation Gaussian Processes 



This research was funded/supported by the Health Innovation Challenge Fund (WT106882), the Wellcome/EPSRC Centre for Medical Engineering [WT203148/Z/16/Z], and the National Institute for Health Research (NIHR) Biomedical Research Centre based at Guy’s and St Thomas’ NHS Foundation Trust and King’s College London and/or the NIHR Clinical Research Facility. We are grateful to the Wolfson Foundation and the Epilepsy Society for supporting the Epilepsy Society MRI scanner. The views expressed in this publication are those of the authors and not necessarily those of the Wellcome Trust, NHS, the NIHR or the Department of Health.


  1. 1.
    Álvarez, M.A., Rosasco, L., Lawrence, N.D.: Kernels for vector-valued functions: a review. J. Found. Trends Mach. Learn. 4(3), 195–266 (2012)CrossRefGoogle Scholar
  2. 2.
    Bhattacharjee, S., Matou\(\check{s}\), K.: A nonlinear manifold-based reduced order model for multiscale analysis of heterogeneous hyperelastic materials. J. Comput. Phys. 313, 635–653 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Budday, S., Sommer, G., Birkl, C., et al.: Mechanical characterization of human brain tissue. Acta Biomater. 48, 319–340 (2017)CrossRefGoogle Scholar
  4. 4.
    Bouaziz, S., Martin, S., Liu, T., et al.: Projective dynamics: fusing constraint projections for fast simulation. ACM Trans. Graph. 33(4), 1–11 (2014)CrossRefGoogle Scholar
  5. 5.
    GPy: A Gaussian Process Framework in Python (2014)Google Scholar
  6. 6.
    Hamhaber, U., Sack, I., Papazoglou, S., et al.: 3D analysis of shear wave propagation observed by in vivo MRE of the brain. Acta Biomater. 3, 127–137 (2007)CrossRefGoogle Scholar
  7. 7.
    Martin, S., Thomaszewski, B., Grispun, E., Gross, M.: Example-based elastic materials. ACM Trans. Graph. 30(4), 1–8 (2011)CrossRefGoogle Scholar
  8. 8.
    Mihai, L.A., Chin, L., Janmey, P.A., Goriely, A.: A comparison of hyperelastic constitutive models applicable to brain and fat tissues. Royal Soc. 12, 1–12 (2015)Google Scholar
  9. 9.
    Mihai, L.A., Budday, S., Holzapfel, G.A., et al.: A family of hyperelastic models for human brain tissue. J. Mech. Phys. Solids 106, 60–79 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mihai, L.A., Wooley, T.E., Goriely, A.: Stochastic isotropic hyperelastic materials: constitutive calibration and model selection. Proc. Royal Soc. A 474, 1–20 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Morin, F., Chabanas, M., Courtecuisse, H., Payan, Y.: Biomechanical modelling of brain soft tissues for medical applications. In: Biomechanics Living Organs, pp. 127–146 (2017)CrossRefGoogle Scholar
  12. 12.
    Schulz, E., Speekenbrink, M., Krause, A.: A tutorial on GP regression: modelling, exploring, and exploiting functions. J. Math. Psychol. 85, 1–16 (2018)CrossRefGoogle Scholar
  13. 13.
    Sifakis, E.D.: FEM Simulation of 3D Deformable Solids: a practitioner’s guide to theory, discretization and model reduction. SIGGRAPH Course, pp. 1–32 (2012)Google Scholar
  14. 14.
    Stomakhin, A., Hower, R., Schroeder, C., Teran, J.M.: Energetically consistent invertible elasticity. In: Eurographics, pp. 1–8 (2012)Google Scholar
  15. 15.
    Teran, J., Sifakis, E., Irving, G., Fedkiw, R.: Robust quasistatic finite elements and flesh simulation. In: EG/SIGGRAPH Symposium on Computer animation, pp. 1–11 (2005)Google Scholar
  16. 16.
    Xu, H., Sin, F., Zhu, Y., Barbic, J.: Nonlinear material design using principal stretches. ACM Trans. Graph. 34(4), 1–11 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alejandro Granados
    • 1
    Email author
  • Martin Schweiger
    • 1
  • Vejay Vakharia
    • 2
  • Andrew W. McEvoy
    • 2
  • Anna Miserocchi
    • 2
  • John S. Duncan
    • 2
    • 3
  • Rachel Sparks
    • 1
  • Sébastien Ourselin
    • 1
  1. 1.School of Biomedical Engineering and Imaging SciencesKing’s College LondonLondonUK
  2. 2.National Hospital of Neurology and NeurosurgeryLondonUK
  3. 3.Department of Clinical and Experimental EpilepsyUCL Queen Square Institute of NeurologyLondonUK

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