Abstract
Computational assistance gains increasing importance in the field of medical surgery. As an example, in the present work, we look at functional endoscopic sinus surgery. Simulations for surgery training programs or online support during surgeries require simulation tools which are characterized by a preferably short simulation time (real time) and a high degree of accuracy. The nonlinear finite element method is most suitable to yield qualitatively and quantitatively reliable results. The problem is, however, to achieve such results in real time. One possibility to reach both, short computational time and high accuracy, is to combine model reduction and finite element techniques. Therefore, in this paper, various projection-based model reduction methods are discussed and compared with respect to their possible application in biomechanics. The modal basis, the load-dependent Ritz and the proper orthogonal decomposition (POD) method were used to reduce the model of a cube under compression considering different material nonlinearities and large deformations. The POD method led to the lowest errors in displacement and stress while providing the largest reduction in CPU time. Further, the influence of different POD parameters was investigated. According to this study, the snapshots upon which the POD is based had to agree as closely as possible with the original deformation of the reduced system. The POD method applied to the finite element model of an inferior turbinate led to an adequate accuracy for surgery simulations within less than one-third of the computational time of the unreduced finite element simulation.
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Amsallem D., Cortial J., Carlberg K., Farhat C.: A method for interpolating on manifolds structural dynamics reduced-order models. Int. J. Numer. Methods Eng. 80(9), 1241–1258 (2009). doi:10.1002/nme.2681
Barbič J., James D.L.: Real-time subspace integration for St. Venant-Kirchhoff deformable models. ACM Trans. Graph. 24(3), 982–990 (2005). doi:10.1145/1073204.1073300
Basdogan, C.: Real-time simulation of dynamically deformable finite element models using modal analysis and spectral Lanczos decomposition methods. Medicine Meets Virtual Reality, pp. 46–52. http://hdl.handle.net/2014/16481 (2001)
Berkley J., Turkiyyah G., Berg D., Ganter M., Weghorst S.: Real-time finite element modeling for surgery simulation: an application to virtual suturing. Vis. Comput. Graph. IEEE Trans. 10(3), 314–325 (2004). doi:10.1109/TVCG.2004.1272730
Bolzon G., Buljak V.: An effective computational tool for parametric studies and identification problems in materials mechanics. Comput. Mech. 48, 675–687 (2011). doi:10.1007/s00466-011-0611-8
Breuer K.S., Sirovich L.: The use of the Karhunen-Loéve procedure for the calculation of linear eigenfunctions. J. Comput. Phys. 96(2), 277–296 (1991). doi:10.1016/0021-9991(91)90237-F
Bro-Nielsen, M.: Finite element modeling in surgery simulation. In: Proceedings of the IEEE, pp. 490–503 (1998)
Bro-Nielsen M., Cotin S.: Real-time volumetric deformable models for surgery simulation using finite elements and condensation. Comput. Graph. Forum 15(3), 57–66 (1996). doi:10.1111/1467-8659.1530057
Carlberg K., Bou-Mosleh C., Farhat C.: Efficient non-linear model reduction via a least-squares Petrov/Galerkin projection and compressive tensor approximations. Int. J. Numer. Methods Eng. 0, 1–25 (2009)
Chatterjee A.: An introduction to the proper orthogonal decomposition. Curr. Sci. 78(7), 808–817 (2000)
Chaturantabut, S.: Nonlinear model reduction via discrete empirical interpolation. PhD thesis, Rice University (2011)
Chaturantabut, S., Sorensen, D.: Discrete empirical interpolation for nonlinear model reduction. In: CDC/CCC 2009. Proceedings of the 48th IEEE Conference on Decision and Control (CDC) and the Chinese Control Conference (CCC), pp. 4316–4321 (2009). doi:10.1109/CDC.2009.5400045
Cotin S., Delingette H., Ayache N.: Real-time elastic deformations of soft tissues for surgery simulation. IEEE Trans. Vis. Comput. Graph. 5, 62–73 (1998). doi:10.1109/2945.764872
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, 437–452 (2000). doi:10.1007/PL00007215
Courtecuisse, H., Jung, H., Allard, J., Duriez, C., Lee, D.Y., Cotin, S.: Gpu-based real-time soft tissue deformation with cutting and haptic feedback. Prog. Biophys. Mol. Biol. 103(23), 159–168 (2010). doi:10.1016/j.pbiomolbio.2010.09.016. <ce:title>Special Issue on Biomechanical Modelling of Soft Tissue Motion</ce:title>
Cover S., Ezquerra N., O’Brien J., Rowe R., Gadacz T., Palm E.: Interactively deformable models for surgery simulation. Comput. Graph. Appl. IEEE 13(6), 68–75 (1993). doi:10.1109/38.252559
De, S., Kim, J., Srinivasan, M.A.: A meshless numerical technique for physically based real-time medical simulations. In: Westwood, J. (ed.) Proceedings of Medicine Meets Virtual Reality, vol. 81, pp. 113–118. IOS Press, Amsterdam (2001)
De S., Deo D., Sankaranarayanan G., Arikatla V.S.: A physics-driven neural networks-based simulation system (phynness) for multimodal interactive virtual environments involving nonlinear deformable objects. Presence Teleoper Virtual Environ. 20(4), 289–308 (2011). doi:10.1162/PRES_a_00054
Debunne, G., Desbrun, M., Cani, M.P., Barr, A.H.: Dynamic real-time deformations using space & time adaptive sampling. In: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’01, pp. 31–36. ACM, New York (2001). doi:10.1145/383259.383262
Delingette H.: Toward realistic soft-tissue modeling in medical simulation. Proc. IEEE 86(3), 512–523 (1998). doi:10.1109/5.662876
Delingette, H.: Biquadratic and quadratic springs for modeling St Venant Kirchhoff materials. In: Bello, F., Edwards, P. (eds.) Biomedical Simulation, Lecture Notes in Computer Science, vol. 5104, pp. 40–48. Springer, Berlin (2008). doi:10.1007/978-3-540-70521-5_5
Delingette, H., Subsol, G., Cotin, S., Pignon, J.M.: A craniofacial surgers simulation testbed. INRIA, Research Report no. 2199, pp. 1–14 (1994)
Dogan F., Serdar Celebi M.: Real-time deformation simulation of non-linear viscoelastic soft tissues. Simulation 87, 179–187 (2011). doi:10.1177/0037549710364532
Edmond C.V.: Impact of the endoscopic sinus surgical simulator on operating room performance. Laryngoscope 112(7), 1148–1158 (2002). doi:10.1097/00005537-200207000-00002
Eichhorn K., Tingelhoff K., Wagner I., Westphal R., Rilk M., Kunkel M., Wahl F., Bootz F.: Sensorbasierte Messung mechanischer Kräfte am Endoskop während FESS. HNO 56(8), 789–794 (2008). doi:10.1007/s00106-007-1647-0
Fukunaga K.: Introduction to Statistical Pattern Recognition. Academic Press, New York (1990)
Gibson, S., Samosky, J., Mor, A., Fyock, C., Grimson, E., Kanade, T., Kikinis, R., Lauer, H., McKenzie, N., Nakajima, S., Ohkami, H., Osborne, R., Sawada, A.: Simulating arthroscopic knee surgery using volumetric object representations, real-time volume rendering and haptic feedback. In: Troccaz, J., Grimson, E., Msges, R. (eds.) CVRMed-MRCAS’97, Lecture Notes in Computer Science, vol. 1205, pp. 367–378. Springer, Berlin (1997). doi:10.1007/BFb0029258
Gibson, S.F.: 3d chainmail: a fast algorithm for deforming volumetric objects. In: Proceedings of the 1997 Symposium on Interactive 3D Graphics, I3D ’97, pp. 149–ff. ACM, New York (1997). doi:10.1145/253284.253324
Holzapfel, G.A.: Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Wiley, Baffins Lane (2000)
Idelsohn S.R., Cardona A.: A reduction method for nonlinear structural dynamic analysis. Comput. Methods Appl. Mech. Eng. 49(3), 253–279 (1985). doi:10.1016/0045-7825(85)90125-2
James, D.L., Pai, D.K.: Artdefo: accurate real time deformable objects. In: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’99, pp. 65–72. ACM Press/Addison-Wesley Publishing Co., New York (1999). doi:10.1145/311535.311542
Joldes, G., Wittek, A., Couton, M., Warfield, S., Miller, K.: Real-time prediction of brain shift using nonlinear finite element algorithms. In: Yang, G.Z., Hawkes, D., Rueckert, D., Noble, A., Taylor, C. (eds.) Medical Image Computing and Computer-Assisted Intervention MICCAI 2009, Lecture Notes in Computer Science, vol. 5762, pp. 300–307. Springer, Berlin (2009). doi:10.1007/978-3-642-04271-3_37
Kapania, R.K., Byun, C.: Reductionmethods based on eigenvectors and Ritz vectors for nonlinear transient analysis.Comput. Mech. 11, 65–82 (1993). doi:10.1007/BF00370072
Kerfriden P., Gosselet P., Adhikari S., Bordas S.: Bridging proper orthogonal decomposition methods and augmented Newton-Krylov algorithms: an adaptive model order reduction for highly nonlinear mechanical problems. Comput. Methods Appl. Mech. Eng. 200(58), 850–866 (2011). doi:10.1016/j.cma.2010.10.009
Kerschen G., Golinval J.C., Vakakis A., Bergman L.: The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dyn. 41(1–3), 147–169 (2005). doi:10.1007/s11071-005-2803-2
Kline K.A.: Dynamic analysis using a reduced basis of exact modes and Ritz vectors. AIAA J. 24(12), 2022–2029 (1986)
Koch, R.M., Gross, M.H., Carls, F.R., von Büren, D.F., Fankhauser, G., Parish, Y.I.H.: Simulating facial surgery using finite element models. In: Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’96, pp. 421–428. ACM, New York (1996). doi:10.1145/237170.237281
Krysl, P., Lall, S., Marsden, J.E.: Dimensional model reduction in nonlinear finite element dynamics of solids and structures. Int. J. Numer. Methods Eng. 51(4), 479–504 (2001). doi:10.1002/nme.167
Kühnapfel U., Cakmak H.K., Maa H.: Endoscopic surgery training using virtual reality and deformable tissue simulation. Comput. Graph. 24, 671–682 (2000)
Lenaerts, V., Kerschen, G., Golinval, J., Chevreuils, C.D.: Proper orthogonal decomposition for model updating of nonlinear mechanical systems. In: GOLINVAL 2001 Mechanical Systems and Signal Processing, pp. 31–43 (2001)
Liang Y., Lee H., Lim S., Lin W., Lee K., Wu C.: Proper orthogonal decomposition and its applications part I: theory. J. Sound Vib. 252(3), 527–544 (2002). doi:10.1006/jsvi.2001.4041
Lim, Y.J., De, S.: Real time simulation of nonlinear tissue response in virtual surgery using the point collocation-based method of finite spheres. Comput. Methods Appl. Mech. Eng. 196(3132), 3011–3024 (2007). doi:10.1016/j.cma.2006.05.015, <ce:title>Computational Bioengineering</ce:title>
Lloyd B., Székely G., Harders M.: Identification of spring parameters for deformable object simulation. Vis. Comput. Graph. IEEE Trans. 13(5), 1081–1094 (2007). doi:10.1109/TVCG.2007.1055
Lloyd, B.A., Kirac, S., Székely, G., Harders, M.: Identification of dynamic mass spring parameters for deformable body simulation. In: Mania, K., Reinhard, E. (eds.) Eurographics 2008—Short Papers, pp. 131–134 (2008)
Loève, M.: Probability theory. The University Series in Higher Mathematics (1963)
Lumley J.L., Holmes P., Berkooz G.: Turbulence, Coherent Structures. Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996)
Mahvash M., Hayward V.: High-fidelity haptic synthesis of contact with deformable bodies. Comput. Graph. Appl. IEEE 24(2), 48–55 (2004). doi:10.1109/MCG.2004.1274061
Meier U., Lpez O., Monserrat C., Juan M., Alcaiz M.: Real-time deformable models for surgery simulation: a survey. Comput. Methods Programs Biomed. 77(3), 183–197 (2005). doi:10.1016/j.cmpb.2004.11.002
Messerklinger, W.: Zur Endoskopietechnik des mittleren Nasenganges. Eur. Arch. Oto-Rhino-Laryngol. 221, 297–305 (1978). doi:10.1007/BF00491466
Meyer M., Matthies H.G.: Efficient model reduction in non-linear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods. Comput. Mech. 31, 179–191 (2003)
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 (2007). doi:10.1002/cnm.887
Miller K., Wittek A., Joldes G.: Biomechanics of the brain for computer-integrated surgery. Biomech. Brain Comput. Integr. Surg. 12(2), 25–37 (2010)
Mollemans, W., Schutyser, F., Nadjmi, N., Suetens, P.: Very fast soft tissue predictions with mass tensor model for maxillofacial surgery planning systems. Int. Congr. Ser. 1281, 491–496 (2005). doi:10.1016/j.ics.2005.03.048, cARS 2005: Computer Assisted Radiology and Surgery
Monserrat C., Meier U., Alcaiz M., Chinesta F., Juan M.: A new approach for the real-time simulation of tissue deformations in surgery simulation. Comput. Methods Programs Biomed. 64(2), 77–85 (2001). doi:10.1016/S0169-2607(00)00093-6
Mourelatos Z.: An efficient crankshaft dynamic analysis using substructuring with Ritz vectors. J. Sound Vib. 238(3), 495–527 (2000). doi:10.1006/jsvi.2000.3208
Natsupakpong S., Glu M.C.C.: Determination of elasticity parameters in lumped element (mass-spring) models of deformable objects. Graph. Models 72(6), 61–73 (2010). doi:10.1016/j.gmod.2010.10.001
Nickell R.: Nonlinear dynamics by mode superposition. Comput. Methods Appl. Mech. Eng. 7(1), 107–129 (1976). doi:10.1016/0045-7825(76)90008-6
Niroomandi S., Alfaro I., Cueto E., Chinesta F.: Real-time deformable models of non-linear tissues by model reduction techniques. Comput. Methods Programs Biomed. 91(3), 223–231 (2008). doi:10.1016/j.cmpb.2008.04.008
Noor A.K., Peters J.M.: Reduced basis technique for nonlinear analysis of structures. AIAA J. 18(4), 455–462 (1980). doi:10.2514/3.50778
Picinbono, G., Delingette, H., Ayache, N.: Non-linear and anisotropic elastic soft tissue models for medical simulation. In: Proceedings of the 2001 IEEE, International Conference on Robotics 8 Automation, Seoul, Korea (2001)
Radermacher, A., Reese, S.: Model reduction for complex continua at the example of modeling soft tissue in the nasal area. In: Markert, B. (ed.) Advances in Extended and Multifield Theories for Continua, Lecture Notes in Applied and Computational Mechanics, vol. 59, pp. 197–217. Springer, Berlin (2011). doi:10.1007/978-3-642-22738-7_10
Reese S.: On a physically stabilized one point finite element formulation for three-dimensional finite elasto-plasticity. Comput. Methods Appl. Mech. Eng. 194(45–47), 4685–4715 (2005). doi:10.1016/j.cma.2004.12.012
Reese, S., Wriggers, P., Reddy, B.: A new locking-free brick element technique for large deformation problems in elasticity. Comput. Struct. 75(3), 291–304 (2000). doi:10.1016/S0045-7949(99)00137-6
Remke, J., Rothert, H.: Eine modale Reduktionsmethode zur geometrisch nichtlinearen statischen und dynamischen Finite-Element-Berechnung. Arch. Appl. Mech. 63, 101–115 (1993). doi:10.1007/BF00788916
Remseth S.: Nonlinear static and dynamic analysis of framed structures. Comput. Struct. 10(6), 879–897 (1979). doi:10.1016/0045-7949(79)90057-9
Rickelt-Rolf, C.: Modellreduktion und Substrukturtechnik zur effizienten Simulation dynamischer, teilgeschädigter Systeme. PhD thesis, Technische Universität Carolo-Wilhelmina zu Braunschweig (2009)
Rudman D.T., Stredney D., Sessanna D., Yagel R., Crawfis R., Heskamp D., Edmond C.V., Wiet G.J.: Functional endoscopic sinus surgery training simulator. Laryngoscope 108(11), 1643–1647 (1998). doi:10.1097/00005537-199811000-00010
Ryckelynck D., Benziane D.M.: Multi-level a priori hyper-reduction of mechanical models involving internal variables. Comput. Methods Appl. Mech. Eng. 199(1720), 1134–1142 (2010). doi:10.1016/j.cma.2009.12.003
Schwartz J.M., Denninger M., Rancourt D., Moisan C., Laurendeau D.: Modelling liver tissue properties using a non-linear visco-elastic model for surgery simulation. Med. Image Anal. 9, 103–112 (2005)
Simo J.: On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput. Methods Appl. Mech. Eng. 60(2), 153–173 (1987). doi:10.1016/0045-7825(87)90107-1
Solyar A., Cuellar H., Sadoughi B., Olson T.R., Fried M.P.: Endoscopic sinus surgery simulator as a teaching tool for anatomy education. Am. J. Surg. 196(1), 120–124 (2008). doi:10.1016/j.amjsurg.2007.06.026
Spiess H., Wriggers P.: Reduction methods for FE analysis in nonlinear structural dynamics. PAMM 5(1), 135–136 (2005). doi:10.1002/pamm.200510048
Stammberger H., Posawetz W.: Clinical review—functional endoscopic sinus surgery—concept, indications and results of the Messerklinger technique. Eur. Arch. Oto-Rhino-Laryngol. 247, 63–76 (1990)
Taylor, R.: Feap—a Finite Element Analysis Program, Version 8.3 User manual. University of California at Berkeley, wwwceberkeleyedu/projects/feap (2011)
Taylor Z., Cheng M., Ourselin S.: High-speed nonlinear finite element analysis for surgical simulation using graphics processing units. Med. Imaging IEEE Trans. 27(5), 650–663 (2008). doi:10.1109/TMI.2007.913112
Terzopoulost D., Platt J., Barr A., Fleischert K.: Elastically deformable models. Comput. Graph. 21, 205–214 (1987)
Volkwein S.: Optimal control of a phase-field model using proper orthogonal decomposition. ZAMM J. Appl. Math. Mech. 81(2), 83–97 (2001). doi:10.1002/1521-4001(200102)81:2<83::AID-ZAMM83>3.0.CO;2-R
Volkwein S., Hepberger A.: Impedance identification by POD model reduction techniques. at-Automatisierungstechnik 56(8), 437–446 (2008)
Wilson E.L.: A new method of dynamic analysis for linear and nonlinear systems. Finite Elem. Anal. Design 1(1), 21–23 (1985). doi:10.1016/0168-874X(85)90004-6
Xu S., Liu X., Zhang H., Hu L.: A nonlinear viscoelastic tensor-mass visual model for surgery simulation. Instrum. Meas. IEEE Trans. 60(1), 14–20 (2011). doi:10.1109/TIM.2010.2065450
Zhang D., Wang T., Liu D., Lin G.: Vascular deformation for vascular interventional surgery simulation. Int. J. Med. Robotics Comput. Assist. Surg. 6(2), 171–177 (2010). doi:10.1002/rcs.302
Zhuang, Y., Canny, J.: Haptic interaction with global deformations. In: Robotics and Automation, 2000. Proceedings. ICRA ’00. IEEE International Conference on, vol. 3, pp. 2428–2433 (2000). doi:10.1109/ROBOT.2000.846391
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Radermacher, A., Reese, S. A comparison of projection-based model reduction concepts in the context of nonlinear biomechanics. Arch Appl Mech 83, 1193–1213 (2013). https://doi.org/10.1007/s00419-013-0742-9
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DOI: https://doi.org/10.1007/s00419-013-0742-9