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Quantifying the uncertainty in a hyperelastic soft tissue model with stochastic parameters. (English) Zbl 1460.74065

Summary: We present a simple open-source semi-intrusive computational method to propagate uncertainties through hyperelastic models of soft tissues. The proposed method is up to two orders of magnitude faster than the standard Monte Carlo method. The material model of interest can be altered by adjusting few lines of (FEniCS) code. The method is able to (1) provide the user with statistical confidence intervals on quantities of practical interest, such as the displacement of a tumour or target site in an organ; (2) quantify the sensitivity of the response of the organ to the associated parameters of the material model. We exercise the approach on the determination of a confidence interval on the motion of a target in the brain. We also show that for the boundary conditions under consideration five parameters of the Ogden-Holzapfel-like model have negligible influence on the displacement of the target zone compared to the three most influential parameters. The benchmark problems and all associated data are made available as supplementary material.

MSC:

74L15 Biomechanical solid mechanics
74B20 Nonlinear elasticity
74S60 Stochastic and other probabilistic methods applied to problems in solid mechanics
92C10 Biomechanics
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[1] Babus̆ka, I.; Baumann, C.; Oden, J., A discontinuous hp finite element method for diffusion problems: 1-d analysis, Comput. Math. Appl., 37, 9, 103-122 (1999) · Zbl 0940.65076
[2] Bordas, S.; Duflot, M., Derivative recovery and a posteriori error estimate for extended finite elements, Comput. Methods Appl. Mech. Eng., 196, 35, 3381-3399 (2007) · Zbl 1173.74401
[3] Bui, H. P.; Tomar, S.; Bordas, S. P.A., Real-time error control for surgical simulation, (Brieu, M.; Bordas, S.; Leriche, E.; Sherwin, S., EUROMECH Colloquium 595, Biomechanics and Computer Assisted Surgery Meets Medical Reality (2017), European Mechanics Society: European Mechanics Society Ecole Centrale Lille, Villeneuve dAscq, France)
[4] Pham, D.; Xu, C.; Prince, J., Current methods in medical image segmentation, Ann. Rev. Biomed. Eng., 2, 1, 315-337 (2000)
[5] Klein, S.; van der Heide, U.; Lips, I.; van Vulpen, M.; Staring, M.; Pluim, J., Automatic segmentation of the prostate in 3d MR images by atlas matching using localized mutual information, Med. Phys., 35, 4, 1407-1417 (2008)
[6] Miller, K., Computational biomechanics for patient-specific applications, Ann. Biomed. Eng., 44, 1, 1-2 (2016)
[7] Viceconti, M.; Bellingeri, L.; Cristofolini, L.; Toni, A., A comparative study on different methods of automatic mesh generation of human femurs, Med. Eng. Phys., 20, 1, 1-10 (1998)
[8] Ferrant, M.; Nabavi, A.; Macq, B.; Jolesz, F. A.; Kikinis, R.; Warfield, S., Registration of 3-d intraoperative Mr images of the brain using a finite-element biomechanical model, IEEE Trans. Med. Imaging, 20, 12, 1384-1397 (2001)
[9] Belytschko, T.; Parimi, C.; Moës, N.; Sukumar, N.; Usui, S., Structured extended finite element methods for solids defined by implicit surfaces, Int. J. Numer. Methods Eng., 56, 4, 609-635 (2003) · Zbl 1038.74041
[10] Nguyen, V.; Rabczuk, T.; Bordas, S.; Duflot, M., Meshless methods: a review and computer implementation aspects, Math. Comput. Simul., 79, 3, 763-813 (2008) · Zbl 1152.74055
[11] Stergiopulos, N.; Young, D.; Rogge, T., Computer simulation of arterial flow with applications to arterial and aortic stenoses, J. Biomech., 25, 12, 1477-1488 (1992)
[12] Cotin, S.; Delingette, H.; Ayache, N., Real-time elastic deformations of soft tissues for surgery simulation, IEEE Trans. Vis. Comput. Gr., 5, 1, 62-73 (1999)
[13] Holzapfel, G. A.; Ogden, R. W., Biomechanical Modelling at the Molecular, Cellular and Tissue Levels (2009), Springer
[14] Payan, Y., Soft Tissue Biomechanical Modeling for Computer Assisted Surgery (2012), Springer
[15] Geris, L.; Gomez-Cabrero, D., Uncertainty in Biology (2016), Springer · Zbl 1325.92003
[16] Miller, K.; Chinzei, K., Mechanical properties of brain tissue in tension, J. Biomech., 35, 4, 483-490 (2002)
[17] Labus, K. M.; Puttlitz, C. M., An anisotropic hyperelastic constitutive model of brain white matter in biaxial tension and structura-mechanical relationships, J. Mech. Behav. Biomed. Mater., 62, 195-208 (2016)
[18] Chatelin, S.; Constantinesco, A.; Willinger, R., Fifty years of brain tissue mechanical testing: from in vitro to in vivo investigations, Biorheology, 47, 255-276 (2010)
[19] Laksari, K.; Shafieian, M.; Darvish, K., Constitutive model for brain tissue under finite compression, J. Biomech., 45, 642-646 (2012)
[20] Angela Mihai, L.; Chin, L.; Janmey, P.; Goriely, A., A comparison of hyperelastic constitutive models applicable to brain and fat tissues, J. R. Soc. Interface, 12, 110 (2015)
[21] de Rooij, R.; Kuhl, E., Constitutive modeling of brain tissue: current perspectives, Appl. Mech. Rev., 68, 1 (2016)
[22] Goriely, A.; Budday, S.; Kuhl, E., Chapter two – neuromechanics: from neurons to brain, Advances in Applied Mechanics, vol. 48, 79-139 (2015), Elsevier
[23] Joldes, G.; 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) · Zbl 1225.92021
[24] Caflisch, C. A., Monte Carlo and quasi-Monte Carlo methods, Acta Numer., 7, 1-49 (1998) · Zbl 0949.65003
[25] Ghanem, R.; Spanos, P., Stochastic Finite Elements: A Spectral Approach (1991), Dover publications: Dover publications Mineola, New York · Zbl 0722.73080
[26] Matthies, G. H., Stochastic finite elements: computational approaches to stochastic partial differential equations, J. Appl. Math. Mech., 88, 849-873 (2008) · Zbl 1158.65009
[27] Giraldi, L.; Liu, D.; Matthies, H. G.; Nouy, A., To be or not to be intrusive? The solution of parametric and stochastic equations—the “plain vanilla” Galerkin case, SIAM J. Sci. Comput., 36, 2720-2744 (2014) · Zbl 1310.65132
[28] Dopico-González, C.; New, A.; Browne, M., Probabilistic finite element analysis of the uncemented hip replacementeffect of femur characteristics and implant design geometry, J. Biomech., 43, 3, 512-520 (2010)
[29] Biehler, J.; Gee, M. W.; Wall, W. A., Towards efficient uncertainty quantification in complex and large-scale biomechanical problems based on a Bayesian multi-fidelity scheme, Biomech. Model. Mechanobiol., 14, 3, 489-513 (2015)
[30] Laz, P. J.; Browne, M., A review of probabilistic analysis in orthopaedic biomechanics, J. Eng. Med., 224, 927-943 (2010)
[31] Plantefève, R.; Peterlik, I.; Courtecuisse, H.; Trivisonne, R.; Radoux, J.; Cotin, S., Atlas-based transfer of boundary conditions for biomechanical simulation, Med. Image Comput. Comput. Assist. Interv., 17, 33-40 (2014)
[32] Risholm, P.; Fedorov, A.; Pursley, J.; Tuncali, K.; Cormack, R.; Wells, W., Probabilistic non-rigid registration of prostate images: modeling and quantify uncertainty, Proceedings of the IEEE International Symposium on Biomedical Imaging, 553-556 (2011)
[33] Mohan, P. S.; Nair, P. B.; Keane, A. J., Stochastic projection schemes for deterministic linear elliptic partial differential equations on random domains, Int. J. Numer. Meth. Eng., 85, 874-895 (2011) · Zbl 1217.76065
[34] Lindgren, F.; Rue, H.; Lindström, J., An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach, J. R. Stat. Soc. Ser. B (Stat. Methodol.), 73, 4, 423-498 (2011) · Zbl 1274.62360
[35] Adler, R. J.; Taylor, J., Random Fields and Geometry (2007), Springer-Verlag · Zbl 1149.60003
[36] Cao, Y.; Hussaini, M. Y.; Zhang, T. A., Exploitation of sensitivity derivates for improving sampling methods, AIAA J., 42, 2, 815-822 (2004)
[37] Liu, Y.; Hussaini, M. Y.; Ökten, G., Optimization of a Monte Carlo variance reduction method based on sensitivity derivatives, Appl. Numer. Math., 72, Complete, 160-171 (2013) · Zbl 1302.65013
[38] P. Hauseux, J.S. Hale, S.P. Bordas, Solving the stochastic burgers equation with a sensitivity derivative-driven Monte Carlo method, figshare10.6084/m9.figshare.3561306; P. Hauseux, J.S. Hale, S.P. Bordas, Solving the stochastic burgers equation with a sensitivity derivative-driven Monte Carlo method, figshare10.6084/m9.figshare.3561306
[39] Hauseux, P.; Hale, J. S.; Bordas, S. P., Accelerating Monte Carlo estimation with derivatives of high-level finite element models, Comput. Methods Appl. Mech. Eng., 318, 917-936 (2017) · Zbl 1439.65006
[40] Alnaes, M.; Blechta, J.; Hake, J.; A., J.; Kehlet, B.; Logg, A.; Richardson, C.; Ring, J.; Rognes, M.; Wells, G., The Fenics project version 1.5, Arch. Numer. Softw., 3, 100 (2015)
[41] Logg, A.; Wells, G., Dolfin: automated finite element computing, ACM Trans. Math. Softw., 37, 2, 20:1-20:28 (2010) · Zbl 1364.65254
[42] Robert, C. P.; Casella, G., Monte Carlo statistical methods, Springer Texts in Statistics (2005), Springer-Verlag New York, Inc.: Springer-Verlag New York, Inc. Secaucus, NJ, USA
[43] Farrell, P. E.; Ham, D. A.; Funke, S. W.; Rognes, M. E., Automated derivation of the adjoint of high-level transient finite element programs, SIAM J. Sci. Comput., 35, 4, C369-C393 (2013) · Zbl 1362.65103
[44] Feinberg, J.; Langtangen, H., Chaospy: an open source tool for designing methods of uncertainty quantification, J. Comput. Sci., 11, 46-57 (2015)
[45] Sobol, I., Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates, Math. Comput. Simul., 55, 1-3, 271-280 (2001) · Zbl 1005.65004
[46] Saltelli, A.; Annoni, P.; Azzini, I.; Campolongo, F.; Ratto, M.; Tarantola, S., Variance based sensitivity analysis of model output. design and estimator for the total sensitivity index, Comput. Phys. Commun., 181, 2, 259-270 (2010) · Zbl 1219.93116
[47] Giles, M., Multilevel Monte Carlo methods, Acta Numer., 24, 259-328 (2015) · Zbl 1316.65010
[48] Sobol’, I. M., Uniformly distributed sequences with an additional uniform property, USRR Comput. Math. Math. Phys., 16, 236-242 (1977) · Zbl 0391.10033
[49] Nowak, W.; Tenkleve, S.; Cirpka, O. A., Efficient computation of linearized cross-covariance and auto-covariance matrices of interdependent quantities, Math. Geol., 35, 1, 53-66 (2003) · Zbl 1302.86028
[50] A. Logg, K. A. Mardal, G. N. Wells (Eds.), Automated Solution of Differential Equations by the Finite Element Method, Vol. 84 of Lecture Notes in Computational Science and Engineering, Springer, 2012. doi:10.1007/978-3-642-23099-8; A. Logg, K. A. Mardal, G. N. Wells (Eds.), Automated Solution of Differential Equations by the Finite Element Method, Vol. 84 of Lecture Notes in Computational Science and Engineering, Springer, 2012. doi:10.1007/978-3-642-23099-8 · Zbl 1247.65105
[51] Holzapfel, G.; Ogden, R., Constitutive modelling of passive myocardium: a structurally based framework for material characterization, Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci., 367, 1902, 3445-3475 (2009) · Zbl 1185.74060
[52] Pérez, F.; Granger, B., IPython: a system for interactive scientific computing, Comput. Sci. Eng., 9, 3, 21-29 (2007)
[53] P. Hauseux, J.S. Hale, S. Cotin, S.P. Bordas, Solving a stochastic hyperelasticity equation with a sensitivity derivative-driven Monte Carlo method,figshare10.6084/m9.figshare.4900298; P. Hauseux, J.S. Hale, S. Cotin, S.P. Bordas, Solving a stochastic hyperelasticity equation with a sensitivity derivative-driven Monte Carlo method,figshare10.6084/m9.figshare.4900298
[54] Valencia, A.; Blas, B.; Ortega, J. H., Modeling of brain shift phenomenon for different craniotomies and solid models, J. Appl. Math., 1-21 (2012) · Zbl 1235.74376
[55] Pezzuto, S.; Ambrosi, D.; Quarteroni, A., An orthotropic activestrain model for the myocardium mechanics and its numerical approximation, Eur. J. Mech. A Solids, 48, 83-96 (2014) · Zbl 1406.74502
[56] Balay, S.; Abhyankar, S.; Adams, M.; Brown, J.; Brune, P.; Buschelman, K.; Dalcin, L.; Eijkhout, V.; Gropp, W.; Kaushik, D.; Knepley, M.; Curfman, L.; Rupp, K.; Smith, B.; Zampini, S.; Zhang, H.; Zhang, H., PETSc Users Manual, Technical Report, ANL-95/11 - Revision 3.7 (2016), Argonne National Laboratory
[57] Falgout, R.; Yang, U., Hypre: a library of high performance preconditioners, (Sloot, P. M.A.; Hoekstra, A. G.; Tan, C. J.K.; Dongarra, J. J., Proceedings of the International Conference on Computational Science. Proceedings of the International Conference on Computational Science, Lecture Notes in Computer Science, vol. 2331 (2002), Springer: Springer Berlin Heidelberg), 632-641 · Zbl 1056.65046
[58] Chamberland, E.; Fortin, A.; Fortin, M., Comparison of the performance of some finite element discretizations for large deformation elasticity problems, Comput. Struct., 88, 1112, 664-673 (2010)
[59] Goriely, A.; Geers, M. G.D.; Holzapfel, G. A.; Jayamohan, J.; Jérusalem, A.; Sivaloganathan, S.; Squier, W.; van Dommelen, J.; Waters, S.; Kuhl, E., Mechanics of the brain: perspectives, challenges, and opportunities, Biomech. Model. Mechanobiol., 14, 5, 931-965 (2015)
[60] Feng, Y.; Okamoto, R. J.; Namani, R.; Genin, G. M.; Bayly, P. V., Measurements of mechanical anisotropy in brain tissue and implications for transversely isotropic material models of white matter, J. Mech. Behav. Biomed. Mater., 23, Supplement C, 117-132 (2013)
[61] Zander, E. K., Tensor Approximation Methods for Stochastic Problems (2012), Shaker
[62] Sudret, B., Global sensitivity analysis using polynomial chaos expansions., Reliab. Eng. Syst. Saf., 93, 7, 964-979 (2008)
[63] Wiener, N., The homogeneous chaos, Am. J. Math., 60, 897-936 (1938) · JFM 64.0887.02
[64] Malliavin, P., Stochastic analysis, Grundlehren dermathematischen Wissenschaften (1997), Springer: Springer Berlin, New York · Zbl 0878.60001
[65] Rappel, H.; Beex, L. A.A.; Bordas, S. P.A., Bayesian inference to identify parameters in viscoelasticity, Mech. TimeDepend. Mater., 1-32 (2017)
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