zbMATH — the first resource for mathematics

Quantifying discretization errors for soft tissue simulation in computer assisted surgery: a preliminary study. (English) Zbl 1443.74259
Summary: Errors in biomechanics simulations arise from modelling and discretization. Modelling errors are due to the choice of the mathematical model whilst discretization errors measure the impact of the choice of the numerical method on the accuracy of the approximated solution to this specific mathematical model. A major source of discretization errors is mesh generation from medical images, that remains one of the major bottlenecks in the development of reliable, accurate, automatic and efficient personalized, clinically-relevant Finite Element (FE) models in biomechanics. The impact of mesh quality and density on the accuracy of the FE solution can be quantified with a posteriori error estimates. Yet, to our knowledge, the relevance of such error estimates for practical biomechanics problems has seldom been addressed, see H. P. Bui et al. [“Bordas real-time error control for surgical simulation”, IEEE Trans. Biomed. Eng. 65, No. 3, 596–607 (2018; doi:10.1109/TBME.2017.2695587)]. In this contribution, we propose an implementation of some a posteriori error estimates to quantify the discretization errors and to optimize the mesh. More precisely, we focus on error estimation for a user-defined quantity of interest with the Dual Weighted Residual (DWR) technique. We test its applicability and relevance in three situations, corresponding to experiments in silicone samples and computations for a tongue and an artery, using a simplified setting, i.e., plane linearized elasticity with contractility of the soft tissue modeled as a pre-stress. Our results demonstrate the feasibility of such methodology to estimate the actual solution errors and to reduce them economically through mesh refinement.
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
74L15 Biomechanical solid mechanics
FEniCS; SyFi
Full Text: DOI
[1] 187ps9-187ps9
[2] Romo, A.; Badel, P.; Duprey, A.; Favre, J.-P.; Avril, S., In vitro analysis of localized aneurysm rupture, J. Biomech., 47, 3, 607-616 (2014)
[3] Schmidt, H.; Galbusera, F.; Rohlmann, A.; Zander, T.; Wilke, H.-J., Effect of multilevel lumbar disc arthroplasty on spine kinematics and facet joint loads in flexion and extension: a finite element analysis, Eur. Spine J., 21, 5, 663-674 (2012)
[4] Perrin, D.; Badel, P.; Orgéas, L.; Geindreau, C.; Dumenil, A.; Albertini, J.-N.; Avril, S., Patient-specific numerical simulation of stent-graft deployment: validation on three clinical cases, J. Biomech., 48, 10, 1868-1875 (2015)
[5] Luboz, V.; Bailet, M.; Boichon Grivot, C.; Rochette, M.; Diot, B.; Bucki, M.; Payan, Y., Personalized modeling for real-time pressure ulcer prevention in sitting posture, J. Tissue Viab., 27, 1, 54-58 (2018)
[6] Buchaillard, S.; Brix, M.; Perrier, P.; Payan, Y., Simulations of the consequences of tongue surgery on tongue mobility: implications for speech production in post-surgery conditions, Int. J. Med. Robot. Comput. Ass. Surg., 3, 3, 252-261 (2007)
[7] Courtecuisse, H.; Allard, J.; Kerfriden, P.; Bordas, S. P.A.; Cotin, S.; Duriez, C., Real-time simulation of contact and cutting of heterogeneous soft-tissues, Med. Image Anal., 18, 2, 394-410 (2014)
[8] Grätsch, T.; Bathe, K.-J., A posteriori error estimation techniques in practical finite element analysis, Comput. Struct., 83, 4, 235-265 (2005)
[9] Bijar, A.; Rohan, P.-Y.; Perrier, P.; Payan, Y., Atlas-based automatic generation of subject-specific finite element tongue meshes, Ann. Biomed. Eng., 44, 1, 16-34 (2016)
[10] Shang, F.; Gan, Y.; Guo, Y., Hexahedral mesh generation via constrained quadrilateralization, PLoS ONE, 12, 5 (2017)
[11] Shepherd, J.; Johnson, C., Hexahedral mesh generation for biomedical models in SCIRun, Eng. Comput., 25, 97-114 (2009)
[12] Bucki, M.; Lobos, C.; Payan, Y.; Hitschfeld, N., Jacobian-based repair method for finite element meshes after registration, Eng. Comput., 27, 3, 285-297 (2011)
[13] Ainsworth, M.; Oden, J. T., A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (2000), Wiley-Interscience, New York · Zbl 1008.65076
[14] Verfürth, R., A posteriori error estimation techniques for finite element methods, Numerical Mathematics and Scientific Computation (2013), Oxford University Press, Oxford · Zbl 1279.65127
[15] Nochetto, R. H.; Siebert, K. G.; Veeser, A., Theory of adaptive finite element methods: an introduction, Multiscale, Nonlinear and Adaptive Approximation, 409-542 (2009), Springer, Berlin · Zbl 1190.65176
[16] Bui, H. P.; Tomar, S.; Courtecuisse, H.; Cotin, S.; Bordas, S. P.A., Real-time error control for surgical simulation, IEEE Trans. Biomed. Eng., 65, 3, 596-607 (2018)
[17] Becker, R.; Rannacher, R., A feed-back approach to error control in finite element methods: basic analysis and examples, East-West J. Numer. Math., 4, 4, 237-264 (1996) · Zbl 0868.65076
[18] Becker, R.; Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10, 1-102 (2001) · Zbl 1105.65349
[19] Automated Solution of Differential Equations by The finite Element Method. The FEniCS book, (Logg, A.; Mardal, K.-A.; Wells, G. N., Lecture Notes in Computational Science and Engineering, 84 (2012), Springer, Heidelberg) · Zbl 1247.65105
[20] Rognes, M. E.; Logg, A., Automated goal-oriented error control I: Stationary variational problems, SIAM J. Sci. Comput., 35, 3, C173-C193 (2013) · Zbl 1276.65071
[21] Cowin, S. C.; Humphrey, J. D., Cardiovascular Soft Tissue Mechanics (2001), Springer
[22] Payan, Y.; Ohayon, J., Biomechanics of living organs: hyperelastic constitutive laws for finite element modeling, Academic Press Series in Biomedical Engineering (2017), Elsevier
[23] Ern, A.; Guermond, J.-L., Theory and Practice of Finite Elements, Applied Mathematical Sciences, 159 (2004), Springer-Verlag, New York
[24] Nochetto, R. H.; Veeser, A.; Verani, M., A safeguarded dual weighted residual method, IMA J. Numer. Anal., 29, 1, 126-140 (2009) · Zbl 1168.65070
[25] Becker, R.; Estecahandy, E.; Trujillo, D., Weighted marking for goal-oriented adaptive finite element methods, SIAM J. Numer. Anal., 49, 6, 2451-2469 (2011) · Zbl 1245.65155
[26] Dörfler, W., A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal., 33, 3, 1106-1124 (1996) · Zbl 0854.65090
[27] Le Floc’h, S.; Ohayon, J.; Tracqui, P.; Finet, G.; Gharib, A. M.; Maurice, R. L.; Cloutier, G.; Pettigrew, R. I., Vulnerable atherosclerotic plaque elasticity reconstruction based on a segmentation-driven optimization procedure using strain measurements: theoretical framework, IEEE Trans. Med. Imaging, 28, 7, 1126-1137 (2009)
[28] Meunier, L.; Chagnon, G.; Favier, D.; Orgéas, L.; Vacher, P., Mechanical experimental characterisation and numerical modelling of an unfilled silicone rubber, Polymer Test., 27, 6, 765-777 (2008)
[29] Gerard, J. M.; Ohayon, J.; Luboz, V.; Perrier, P.; Payan, Y., Non-linear elastic properties of the lingual and facial tissues assessed by indentation technique: Application to the biomechanics of speech production, Med. Eng. Phys., 27, 10, 884-892 (2005)
[30] Tracqui, P.; Ohayon, J., Transmission of mechanical stresses within the cytoskeleton of adherent cells: a theoretical analysis based on a multi-component cell model, Acta Biotheor., 52, 4, 323-341 (2004)
[31] Jin, Y.; González-Estrada, O.; Pierard, O.; Bordas, S. P.A., Error-controlled adaptive extended finite element method for 3D linear elastic crack propagation, Comput. Methods Appl. Mech. Eng., 318, 319-348 (2017)
[32] Zienkiewicz, O. C.; Zhu, J. Z., A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Numer. Methods Eng., 24, 2, 337-357 (1987) · Zbl 0602.73063
[33] González-Estrada, O. A.; Nadal, E.; Ródenas, J.; Kerfriden, P.; Bordas, S. P.A.; Fuenmayor, F., Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery, Comput. Mech., 53, 5, 957-976 (2014) · Zbl 1398.74332
[34] González-Estrada, O. A.; Ródenas, J. J.; Bordas, S. P.A.; Nadal, E.; Kerfriden, P.; Fuenmayor, F. J., Locally equilibrated stress recovery for goal oriented error estimation in the extended finite element method, Comput. Struct., 152, 1-10 (2015)
[35] Rüter, M.; Gerasimov, T.; Stein, E., Goal-oriented explicit residual-type error estimates in XFEM, Comput. Mech., 52, 2, 361-376 (2013) · Zbl 1398.74397
[36] Wick, T., Goal functional evaluations for phase-field fracture using PU-based DWR mesh adaptivity, Comput. Mech., 57, 6, 1017-1035 (2016) · Zbl 1382.74130
[37] Giles, M. B.; Süli, E., Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, Acta Numer., 11, 145-236 (2002) · Zbl 1105.65350
[38] Ainsworth, M.; Rankin, R., Guaranteed computable bounds on quantities of interest in finite element computations, Int. J. Numer. Methods Eng., 89, 13, 1605-1634 (2012) · Zbl 1242.65232
[39] Carstensen, C., Estimation of higher Sobolev norm from lower order approximation, SIAM J. Numer. Anal., 42, 5, 2136-2147 (2005) · Zbl 1081.65108
[40] Bangerth, W.; Rannacher, R., Adaptive finite element methods for differential equations, Lectures in Mathematics ETH Zürich (2003), Birkhäuser Verlag, Basel · Zbl 1020.65058
[41] B. Endtmayer, U. Langer, T. Wick, Two-side a posteriori error estimates for the DWR method, arXiv:1811.07586 (2018).
[42] Larsson, F.; Hansbo, P.; Runesson, K., Strategies for computing goal-oriented a posteriori error measures in non-linear elasticity, Int. J. Numer. Methods Eng., 55, 8, 879-894 (2002) · Zbl 1024.74041
[43] Whiteley, J. P.; Tavener, S. J., Error estimation and adaptivity for incompressible hyperelasticity, Int. J. Numer. Methods Eng., 99, 5, 313-332 (2014) · Zbl 1352.74446
[44] Repin, S.; Sauter, S.; Smolianski, A., A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions, Computing, 70, 3, 205-233 (2003) · Zbl 1128.35319
[45] Oden, J. T.; Prudhomme, S., Estimation of modeling error in computational mechanics, J. Comput. Phys., 182, 2, 496-515 (2002) · Zbl 1053.74049
[46] Braack, M.; Ern, A., A posteriori control of modeling errors and discretization errors, SIAM J. Multiscale Model. Simul., 1, 2, 221-238 (2003) · Zbl 1050.65100
[47] (to appear).
[48] Rappel, H.; Beex, L. A.; Bordas, S. P.A., Bayesian inference to identify parameters in viscoelasticity, Mech. Time Depend. Mater., 22, 2, 251-258 (2018)
[49] Moireau, P.; Chapelle, D.; Le Tallec, P., Filtering for distributed mechanical systems using position measurements: perspectives in medical imaging, Inverse Probl., 25, 3, 035010, 25 (2009) · Zbl 1169.35393
[50] Haouchine, N.; Dequidt, J.; Peterlik, I.; Kerrien, E.; Berger, M.-O.; Cotin, S., Image-guided simulation of heterogeneous tissue deformation for augmented reality during hepatic surgery, Proceedings of the IEEE International Symposium on Mixed and Augmented Reality (ISMAR), 199-208 (2013), IEEE
[51] Hauseux, P.; Hale, J. S.; Bordas, S. P.A., Accelerating Monte Carlo estimation with derivatives of high-level finite element models, Comput. Methods Appl. Mech. Eng., 318, 917-936 (2017)
[52] Becker, R.; Vexler, B., Mesh refinement and numerical sensitivity analysis for parameter calibration of partial differential equations, J. Comput. Phys., 206, 1, 95-110 (2005) · Zbl 1082.65130
[53] Eigel, M.; Merdon, C.; Neumann, J., An adaptive multilevel Monte Carlo method with stochastic bounds for quantities of interest with uncertain data, SIAM/ASA J. Uncerta. Quant., 4, 1, 1219-1245 (2016) · Zbl 1398.35306
[54] Guignard, D.; Nobile, F.; Picasso, M., A posteriori error estimation for elliptic partial differential equations with small uncertainties, Numer. Methods Partial Differ. Equ., 32, 1, 175-212 (2016) · Zbl 1350.65007
[55] Akbari Rahimabadi, A.; Kerfriden, P.; Bordas, S. P.A., Scale selection in nonlinear fracture mechanics of heterogeneous materials, Philosoph. Mag., 95, 28-30, 3328-3347 (2015)
[56] 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)
[57] Niroomandi, S.; Alfaro, I.; Gonzalez, D.; Cueto, E.; Chinesta, F., Real-time simulation of surgery by reduced-order modeling and X-FEM techniques, Int. J. Numer. Methods Biomed. Eng., 28, 5, 574-588 (2012) · Zbl 1243.92032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.