Skip to main content

Physics-Informed Neural Networks for Tissue Elasticity Reconstruction in Magnetic Resonance Elastography

  • Conference paper
  • First Online:
Medical Image Computing and Computer Assisted Intervention – MICCAI 2023 (MICCAI 2023)

Abstract

Magnetic resonance elastography (MRE) is a medical imaging modality that non-invasively quantifies tissue stiffness (elasticity) and is commonly used for diagnosing liver fibrosis. Constructing an elasticity map of tissue requires solving an inverse problem involving a partial differential equation (PDE). Current numerical techniques to solve the inverse problem are noise-sensitive and require explicit specification of physical relationships. In this work, we apply physics-informed neural networks to solve the inverse problem of tissue elasticity reconstruction. Our method does not rely on numerical differentiation and can be extended to learn relevant correlations from anatomical images while respecting physical constraints. We evaluate our approach on simulated data and in vivo data from a cohort of patients with non-alcoholic fatty liver disease (NAFLD). Compared to numerical baselines, our method is more robust to noise and more accurate on realistic data, and its performance is further enhanced by incorporating anatomical information.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Manduca, A., et al.: Magnetic resonance elastography: non-invasive mapping of tissue elasticity. Med. Image Anal. 5(4), 237–254 (2001). https://doi.org/10.1016/s1361-8415(00)00039-6

    Article  Google Scholar 

  2. Petitclerc, L., Sebastiani, G., Gilbert, G., Cloutier, G., Tang, A.: Liver fibrosis: review of current imaging and MRI quantification techniques. J. Magn. Reson. Imaging 45(5), 1276–1295 (2016)

    Article  Google Scholar 

  3. Oliphant, T.E., Manduca, A., Ehman, R.L., Greenleaf, J.F.: Complex-valued stiffness reconstruction for magnetic resonance elastography by algebraic inversion of the differential equation. Magn. Reson. Med. 45(2), 299–310 (2001). https://doi.org/10.1002/1522-2594(200102)45:2<299::aid-mrm1039>3.0.co;2-o

    Article  Google Scholar 

  4. Park, E., Maniatty, A.M.: Shear modulus reconstruction in dynamic elastography: time harmonic case. Phys. Med. Biol. 51, 3697 (2006). https://doi.org/10.1088/0031-9155/51/15/007

    Article  Google Scholar 

  5. Papazoglou, S., Hamhaber, U., Braun, J., Sack, I.: Algebraic Helmholtz inversion in planar magnetic resonance elastography. Phys. Med. Biol. 53(12), 3147–3158 (2008). https://doi.org/10.1088/0031-9155/53/12/005

    Article  Google Scholar 

  6. Eskandari, H., Salcudean, S.E., Rohling, R., Bell, I.: Real-time solution of the finite element inverse problem of viscoelasticity. Inverse Prob. 27(8), 085002 (2011). https://doi.org/10.1088/0266-5611/27/8/085002

    Article  MathSciNet  MATH  Google Scholar 

  7. Honarvar, M., Sahebjavaher, R., Sinkus, R., Rohling, R., Salcudean, S.E.: Curl-based finite element reconstruction of the shear modulus without assuming local homogeneity: Time harmonic case. IEEE Tran. Med. Imaging 32(12), 2189–99 (2013). https://doi.org/10.1109/TMI.2013.2276060

    Article  Google Scholar 

  8. Honarvar, M., Rohling, R., Salcudean, S.E.: A comparison of direct and iterative finite element inversion techniques in dynamic elastography. Phys. Med. Biol. 61(8), 3026–48 (2016). https://doi.org/10.1088/0031-9155/61/8/3026

    Article  Google Scholar 

  9. Fovargue, D., Nordsletten, D., Sinkus, R.: Stiffness reconstruction methods for MR elastography. NMR Biomed. 31(10), e3935 (2018). https://doi.org/10.1002/nbm.3935

    Article  Google Scholar 

  10. Fovargue, D., Kozerke, S., Sinkus, R., Nordsletten, D.: Robust MR elastography stiffness quantification using a localized divergence free finite element reconstruction. Med. Image Anal. 44, 126–142 (2018)

    Article  Google Scholar 

  11. Murphy, M.C., Manduca, A., Trzasko, J.D., Glaser, K.J., Huston III, J., Ehman, R.L.: Artificial neural networks for stiffness estimation in magnetic resonance elastography. Magn. Reson. Med. 80(1), 351–360 (2017)

    Google Scholar 

  12. Solamen, L., Shi, Y., Amoh, J.: Dual objective approach using a convolutional neural network for magnetic resonance elastography. arXiv preprint: 1812.00441 [physics.med-ph] (2018)

    Google Scholar 

  13. Ni, B., Gao, H.: A deep learning approach to the inverse problem of modulus identification in elasticity. MRS Bull. 46(1), 19–25 (2021). https://doi.org/10.1557/s43577-020-00006-y

    Article  Google Scholar 

  14. Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707 (2019). https://doi.org/10.1016/j.jcp.2018.10.045

    Article  MathSciNet  MATH  Google Scholar 

  15. Haghighat, E., Raissi, M., Moure, A., Gomez, H., Juanes, R.: A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics. Comput. Methods Appl. Mech. Eng., 113741 (2021). https://doi.org/10.1016/j.cma.2021.113741

  16. Zhang, E., Yin, M., Karniadakis, G.E.: Physics-informed neural networks for nonhomogeneous material identification in elasticity imaging. arXiv preprint: 2009.04525 [cs.LG] (2020). https://doi.org/10.48550/arXiv.2009.04525

  17. Mallampati, A., Almekkawy, M.: Measuring tissue elastic properties using physics based neural networks. In: 2021 IEEE UFFC Latin America Ultrasonics Symposium (LAUS), pp. 1–4. IEEE, Gainesville (2021). https://doi.org/10.1109/LAUS53676.2021.9639231

  18. Kamali, A., Sarabian, M., Laksari, K.: Elasticity imaging using physics-informed neural networks: spatial discovery of elastic modulus and Poisson’s ratio. Acta Biomater. 155, 400–409 (2023). https://doi.org/10.1016/j.actbio.2022.11.024

    Article  Google Scholar 

  19. Wymer, D.T., Patel, K.P., Burke, W.F., III., Bhatia, V.K.: Phase-contrast MRI: physics, techniques, and clinical applications. RadioGraphics 40(1), 122–140 (2020)

    Article  Google Scholar 

  20. Sinkus, R., Daire, J.L., Beers, B.E.V., Vilgrain, V.: Elasticity reconstruction: beyond the assumption of local homogeneity. Comptes Rendus Mécanique 338(7), 474–479 (2010). https://doi.org/10.1016/j.crme.2010.07.014

    Article  Google Scholar 

  21. Honarvar, M.: Dynamic elastography with finite element-based inversion. Ph.D. thesis, University of British Columbia (2015). https://doi.org/10.14288/1.0167683

  22. Pollack, B.L., et al.: Deep learning prediction of voxel-level liver stiffness in patients with nonalcoholic fatty liver disease. Radiology: AI 3(6) (2021). https://doi.org/10.1148/ryai.2021200274

  23. Sitzmann, V., Martel, J.N.P., Bergman, A.W., Lindell, D.B., Wetzstein, G.: Implicit neural representations with periodic activation functions (2020)

    Google Scholar 

  24. Kingma, D.P., Ba, J.L.: Adam: a method for stochastic optimization. In: Proceedings of 3rd International Conference Learning Representations (2015)

    Google Scholar 

  25. Paszke, A., et al.: PyTorch: an imperative style, high-performance deep learning library. In: Advance Neural Information Processing System, vol. 32, pp. 8024–8035. Curran Associates, Inc. (2019)

    Google Scholar 

  26. Lu, L., Meng, X., Mao, Z., Karniadakis, G.E.: DeepXDE: a deep learning library for solving differential equations. SIAM Rev. 63(1), 208–228 (2021). https://doi.org/10.1137/19M1274067

    Article  MathSciNet  MATH  Google Scholar 

  27. Scroggs, M.W., Dokken, J.S., Richardson, C.N., Wells, G.N.: Construction of arbitrary order finite element degree-of-freedom maps on polygonal and polyhedral cell meshes. ACM Trans. Math. Softw. 48, 1–23 (2022). https://doi.org/10.1145/3524456

    Article  MathSciNet  MATH  Google Scholar 

  28. Barnhill, E., Davies, P.J., Ariyurek, C., Fehlner, A., Braun, J., Sack, I.: Heterogeneous multifrequency direct inversion (HMDI) for magnetic resonance elastography with application to a clinical brain exam. Med. Image Anal. 46, 180–188 (2018). https://doi.org/10.1016/j.media.2018.03.003

    Article  Google Scholar 

  29. Kallel, F., Bertrand, M., Ophir, J.: Fundamental limitations on the contrast-transfer efficiency in elastography: an analytic study. Ultrasound Med. Biol. 22(4), 463–470 (1996). https://doi.org/10.1016/0301-5629(95)02079-9

    Article  Google Scholar 

  30. Lowekamp, B.C., Chen, D.T., Ibáñez, L., Blezek, D.: The design of SimpleITK. Front. Neuroinf. 7(45) (2013). https://doi.org/10.3389/fninf.2013.00045

  31. Mueller, S., Sandrin, L.: Liver stiffness: a novel parameter for the diagnosis of liver disease. Hepat. Med. 2, 49–67 (2010). https://doi.org/10.2147/hmer.s7394

    Article  Google Scholar 

Download references

Acknowledgements & Data Use

This work was supported by the Pennsylvania Department of Health (grant number 41000873310), National Institutes of Health (grant number R01HL141813), the National Science Foundation (grant number 1839332) and Tripod+X. This work used the Bridges-2 system, which is supported by NSF award number OAC-1928147 at the Pittsburgh Supercomputing Center (PSC).

The patient MRE data was acquired by Amir A. Borhani, MD while he was at University of Pittsburgh. We thank him for his collaboration and guidance during this project.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Matthew Ragoza .

Editor information

Editors and Affiliations

1 Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 141 KB)

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Ragoza, M., Batmanghelich, K. (2023). Physics-Informed Neural Networks for Tissue Elasticity Reconstruction in Magnetic Resonance Elastography. In: Greenspan, H., et al. Medical Image Computing and Computer Assisted Intervention – MICCAI 2023. MICCAI 2023. Lecture Notes in Computer Science, vol 14229. Springer, Cham. https://doi.org/10.1007/978-3-031-43999-5_32

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-43999-5_32

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-43998-8

  • Online ISBN: 978-3-031-43999-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics