From computed tomography to finite element space: A unified bone material mapping strategy

  • Petr Henyš
    Institute of New Technologies and Applied Informatics, Faculty of Mechatronics, Informatics and Interdisciplinary Studies, Technical University of Liberec, Studentská 1402/2, 46117 Liberec, Czech Republic
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  • Miroslav Vořechovský
    Institute of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, 60200 Brno, Czech Republic
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  • Jan Stebel
    Institute of New Technologies and Applied Informatics, Faculty of Mechatronics, Informatics and Interdisciplinary Studies, Technical University of Liberec, Studentská 1402/2, 46117 Liberec, Czech Republic
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  • Michal Kuchař
    Department of Anatomy, Faculty of Medicine in Hradec Králové, Charles University, Šimkova 870, 50003, Hradec Králové, Czech Republic
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  • Pavel Exner
    Corresponding author.
    Institute of New Technologies and Applied Informatics, Faculty of Mechatronics, Informatics and Interdisciplinary Studies, Technical University of Liberec, Studentská 1402/2, 46117 Liberec, Czech Republic
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      • Unified least-squares-based framework for fast “nodal” and “element” strategies of mapping scalar CT voxel densities onto finite element mesh.
      • Approximation of density by finite element method using least squares projection.
      • The discontinuous variant of the finite element method is more suitable for density approximation than the continuous variant in terms of accuracy and efficiency.
      • The discontinuous zero-order finite element method preserves the density spectrum better than the continuous one.



      The spatially varying mechanical properties in finite element models of bone are most often derived from bone density data obtained via quantitative computed tomography. The key step is to accurately and efficiently map the density given in voxels to the finite element mesh.


      The density projection is first formulated in least-squares terms and then discretized using a continuous and discontinuous variant of the finite element method. Both discretization variants are compared with the nodal and element approaches known from the literature.


      In terms of accuracy in the L2 norm, energy distance and efficiency, the discontinuous zero-order variant appears to be the most advantageous. The proposed variant sufficiently preserves the spectrum of density at the edges, while keeping computational cost low.


      The continuous finite element method is analogous to the nodal formulation in the literature, while the discontinuous finite element method is analogous to the element formulation. The two variants differ in terms of implementation, computational cost and ability to preserve the density spectrum. These differences cannot be described and measured by known indirect methods from the literature.


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        • Prado M.
        • Khosla S.
        • Chaput C.
        • Giambini H.
        Opportunistic application of phantom-less calibration methods for fracture risk prediction using QCT/FEA.
        Eur. Radiol. 2021; 31: 9428-9435
        • Justin J.T.
        • Smith A.C.
        • Kuczynski M.T.
        • Kaketsis D.A.
        • Manske S.L.
        Advancements in osteoporosis imaging, screening, and study of disease etiology.
        Curr. Osteoporosis Rep. 2021; 19: 532-541
        • Helgason B.
        • Gilchrist S.
        • Ariza O.
        • Vogt P.
        • Enns-Bray W.
        • Widmer R.
        • Fitze T.
        • Pálsson H.
        • Pauchard Y.
        • Guy P.
        • et al.
        The influence of the modulus-density relationship and the material mapping method on the simulated mechanical response of the proximal femur in side-ways fall loading configuration.
        Med. Eng. Phys. 2016; 38: 679-689
        • Michalski A.S.
        • Besler B.A.
        • Michalak G.J.
        • Boyd S.K.
        CT-based internal density calibration for opportunistic skeletal assessment using abdominal CT scans.
        Med. Eng. Phys. 2020; 78 (ISSN 1350-4533): 55-63
        • Fleps I.
        • Bahaloo H.
        • Zysset P.K.
        • Ferguson S.J.
        • Pálsson H.P.
        • Helgason B.
        Empirical relationships between bone density and ultimate strength: a literature review.
        J. Mech. Behavior Biomed. Mat. 2020; 110 (ISSN 1751-6161): 103866
        • Wu S.
        • Todo M.
        • Umebayashi D.
        • Yamamoto Y.
        Risk assessment of vertebral compressive fracture using bone mass index and strength predicted by computed tomography image based finite element analysis.
        Clin. Biomech. 2021; 85 (ISSN 0268-0033): 105365
        • Yadav R.N.
        • Sihota P.
        • Uniyal P.
        • Neradi D.
        • Bose J.C.
        • Dhiman V.
        • Karn S.
        • Sharma S.
        • Aggarwal S.
        • Goni V.G.
        • Kumar S.
        • Kumar Bhadada S.
        • Kumar N.
        Prediction of mechanical properties of trabecular bone in patients with type 2 diabetes using damage based finite element method.
        J. Biomech. 2021; 123 (ISSN 0021-9290): 110495
        • Taylor M.
        • Viceconti M.
        • Bhattacharya P.
        • Li X.
        Finite element analysis informed variable selection for femoral fracture risk prediction.
        J. Mech. Behavior Biomed. Mat. 2021; 118 (ISSN 1751-6161): 104434
        • Lewis G.S.
        • Mischler D.
        • Wee H.
        • Reid J.S.
        • Varga P.
        Finite element analysis of fracture fixation.
        Curr. Osteoporosis Rep. 2021; 19: 403-416
        • Kwak D.-K.
        • Bang S.-H.
        • Kim W.-H.
        • Lee S.-J.
        • Lee S.
        • Yoo J.-H.
        Biomechanics of subtrochanteric fracture fixation using short cephalomedullary nails: a finite element analysis.
        PLOS One. 2021; 16: 1-15
        • Taddei F.
        • Pancanti A.
        • Viceconti M.
        An improved method for the automatic mapping of computed tomography numbers onto finite element models.
        Med. Eng. Phys. 2004; 26 (ISSN 1350-4533): 61-69
        • Helgason B.
        • Taddei F.
        • Pálsson H.
        • Schileo E.
        • Cristofolini L.
        • Viceconti M.
        • Brynjólfsson S.
        A modified method for assigning material properties to FE models of bones, medical engineering.
        Physics. 2008; 30 (ISSN 1350-4533): 444-453
        • Chen G.
        • Schmutz B.
        • Epari D.
        • Rathnayaka K.
        • Ibrahim S.
        • Schuetz M.
        • Pearcy M.
        A new approach for assigning bone material properties from CT images into finite element models.
        J. Biomech. 2010; 43 (ISSN 0021-9290): 1011-1015
        • Chen G.
        • Wu F.
        • Liu Z.
        • Yang K.
        • Cui F.
        Comparisons of node-based and element-based approaches of assigning bone material properties onto subject-specific finite element models.
        Med. Eng. Phys. 2015; 37 (ISSN 1350-4533): 808-812
        • Pegg E.C.
        • Gill H.S.
        An open source software tool to assign the material properties of bone for ABAQUS finite element simulations.
        J. Biomech. 2016; 49 (ISSN 0021-9290): 3116-3121
        • Kalajahi S.M.H.
        • Nazemi S.M.
        • Johnston J.D.
        An exclusion approach for addressing partial volume artifacts with quantitative computed tomography-based finite element modeling of the proximal tibia.
        Med. Eng. Phys. 2020; 76 (ISSN 1350-4533): 95-100
        • Naseri A.B.
        • Dunbar N.J.
        • Baines A.J.
        • Akin J.E.
        • III C.H.
        • Fregly B.J.
        Heterogeneous material mapping methods for patient-specific finite element models of pelvic trabecular bone: a convergence study.
        Med. Eng. Phys. 2021; 96 (ISSN 1350-4533): 1-12
        • Bentley J.L.
        Multidimensional binary search trees used for associative searching.
        CACM. 1975; 18: 509-517
        • Logg A.
        • Mardal K.-A.
        • Wells G.
        Automated Solution of Differential Equations by the Finite Element Method: The FEniC book. 84. Springer Science & Business Media, Heidelberg2012
        • Geuzaine C.
        • Remacle J.-F.
        Gmsh, urlprefix.
        • Rebay S.
        Efficient unstructured mesh generation by means of delaunay triangulation and Bowyer-Watson algorithm.
        J. Comput. Phys. 1993; 106 (ISSN 0021-9991): 125-138
        • Székely G.J.
        • Rizzo M.L.
        Energy statistics: a class of statistics based on distances.
        J. Stat. Plan. Inference. 2013; 143: 1249-1272
        • Virtanen P.
        • Gommers R.
        • Oliphant T.E.
        • Haberland M.
        • Reddy T.
        • Cournapeau D.
        • Burovski E.
        • Peterson P.
        • Weckesser W.
        • Bright J.
        • et al.
        SciPy 1.0 contributors, scipy 1.0: fundamental algorithms for scientific computing in Python.
        Nat. Methods. 2020; 17: 261-272
        • Fat D.L.
        • Kennedy J.
        • Galvin R.
        • O’Brien F.
        • Mc Grath F.
        • Mullett H.
        The Hounsfield value for cortical bone geometry in the proximal humerus-an in vitro study.
        Skeletal Radiol. 2012; 41: 557-568
        • Aamodt A.
        • Kvistad K.A.
        • Andersen E.
        • Lund-Larsen J.
        • Eine J.
        • Benum P.
        • Husby O.S.
        Determination of the Hounsfield value for CT-based design of custom femoral stems.
        J. Bone Joint Surgery. Brit. Volume. 1999; 81-B: 143-147
        • Bank R.E.
        • Yserentant H.
        On the H1-stability of the L2-projection onto finite element spaces.
        Numer. Math. 2014; 126 (ISSN 0029-599X 0945-3245): 361-381
        • Maas S.A.
        • Ellis B.J.
        • Ateshian G.A.
        • Weiss J.A.
        FEBio: finite elements for biomechanics.
        J. Biomech. Eng. 2001; 134 (011005 (10 pages))
        • Schroeder W.J.
        • Avila L.S.
        • Hoffman W.
        Visualizing with VTK: a tutorial.
        IEEE Comput. Graph. Appl. 2000; 20 (ISSN 0272-1716 1558-1756): 20-27
        • Folk M.
        • Heber G.
        • Koziol Q.
        • Pourmal E.
        • Robinson D.
        An overview of the HDF5 technology suite and its applications.
        in: Proceedings of the EDBT/ICDT 2011 Workshop on Array Databases, AD ‘11, Association for Computing Machinery, New York, NY USA2011: 36-47 (2011. ISBN 9781450306140)
        • Lewandowski K.
        • Kaczmarczyk L.
        • Athanasiadis I.
        • Marshall J.F.
        • Pearce C.J.
        A computational framework for crack propagation in spatially heterogeneous materials.
        Philosoph. Trans. Royal Soc. A. 2021; 379: 20200291
        • Ashjaee N.
        • Kalajahi S.M.H.
        • Johnston J.D.
        QCT-FE modeling of the proximal tibia: effect of mapping strategy on convergence time and model accuracy.
        Med. Eng. Phys. 2021; 88 (ISSN 1350-4533): 41-46
        • Arnold D.N.
        • Logg A.
        Periodic table of the finite elements.
        Siam News. 2014; 47: 212