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Patient-Specific Biomechanical Modeling of Bone Strength Using Statistically-Derived Fabric Tensors

  • Computational Biomechanics for Patient-Specific Applications
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Abstract

Low trauma fractures are amongst the most frequently encountered problems in the clinical assessment and treatment of bones, with dramatic health consequences for individuals and high financial costs for health systems. Consequently, significant research efforts have been dedicated to the development of accurate computational models of bone biomechanics and strength. However, the estimation of the fabric tensors, which describe the microarchitecture of the bone, has proven to be challenging using in vivo imaging. On the other hand, existing research has shown that isotropic models do not produce accurate predictions of stress states within the bone, as the material properties of the trabecular bone are anisotropic. In this paper, we present the first biomechanical study that uses statistically-derived fabric tensors for the estimation of bone strength in order to obtain patient-specific results. We integrate a statistical predictive model of trabecular bone microarchitecture previously constructed from a sample of ex vivo micro-CT datasets within a biomechanical simulation workflow. We assess the accuracy and flexibility of the statistical approach by estimating fracture load for two different databases and bone sites, i.e., for the femur and the T12 vertebra. The results obtained demonstrate good agreement between the statistically-driven and micro-CT-based estimates, with concordance coefficients of 98.6 and 95.5% for the femur and vertebra datasets, respectively.

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References

  1. Abdi, H. Partial least squares regression (PLS-regression). Thousand Oaks: Sage, pp. 792–795, 2003.

    Google Scholar 

  2. Åkesson, K., D. Marsh, P. Mitchell, A. McLellan, J. Stenmark, D. Pierroz, C. Kyer, C. Cooper, and I. F. W. Group. Capture the fracture: a best practice framework and global campaign to break the fragility fracture cycle. Osteoporos. Int. 24:2135–2152, 2013.

    Article  PubMed  PubMed Central  Google Scholar 

  3. Arlot, M. E., B. Burt-Pichat, J. P. Roux, D. Vashishth, M. L. Bouxsein, and P. D. Delmas. Microarchitecture influences microdamage accumulation in human vertebral trabecular bone. J. Bone Miner. Res. 23:1613–1618, 2008.

    Article  PubMed  PubMed Central  Google Scholar 

  4. Bayraktar, H. H., E. F. Morgan, G. L. Niebur, G. E. Morris, E. K. Wong, and T. M. Keaveny. Comparison of the elastic and yield properties of human femoral trabecular and cortical bone tissue. J. Biomech. 37:27–35, 2004.

    Article  PubMed  Google Scholar 

  5. Bookstein, F. L. Principal warps: thin-plate splines and the decomposition of deformations. IEEE Trans. Pattern Anal. Mach. Intell. 11:567–585, 1989.

    Article  Google Scholar 

  6. Boutroy, S., M. L. Bouxsein, F. Munoz, and P. D. Delmas. In vivo assessment of trabecular bone microarchitecture by high-resolution peripheral quantitative computed tomography. J. Clin. Endocrinol. Metab. 90:6508–6515, 2005.

    Article  PubMed  CAS  Google Scholar 

  7. Brandi, M. L. Microarchitecture, the key to bone quality. Rheumatology 48:iv3–iv8, 2009.

    Article  PubMed  Google Scholar 

  8. Burrows, M., D. Liu, and H. McKay. High-resolution peripheral QCT imaging of bone micro-structure in adolescents. Osteoporos. Int. 21:515–520, 2010.

    Article  PubMed  CAS  Google Scholar 

  9. Capture the fracture. Report 2012. International Osteoporosis Foundation, 2012.

  10. Chappard, D., M.-F. Baslé, E. Legrand, and M. Audran. Trabecular bone microarchitecture: a review. Morphologie 92:162–170, 2008.

    Article  PubMed  CAS  Google Scholar 

  11. Charlebois, M., M. Jirásek, and P. K. Zysset. A nonlocal constitutive model for trabecular bone softening in compression. Biomech. Model. Mechanobiol. 9:597–611, 2010.

    Article  PubMed  Google Scholar 

  12. Cowin, S. C. The relationship between the elasticity tensor and the fabric tensor. Mech. Mater. 4:137–147, 1985.

    Article  Google Scholar 

  13. Cowin, S. Wolff’s law of trabecular architecture at remodeling equilibrium. J. Biomech. Eng. 108:83–88, 1986.

    Article  PubMed  CAS  Google Scholar 

  14. Cummings, S. R., and L. J. Melton. Epidemiology and outcomes of osteoporotic fractures. The Lancet 359:1761–1767, 2002.

    Article  Google Scholar 

  15. Dalle Carbonare, L., and S. Giannini. Bone microarchitecture as an important determinant of bone strength. J. Endocrinol. Invest. 27:99–105, 2004.

    Article  PubMed  CAS  Google Scholar 

  16. Derikx, L. C., N. Verdonschot, and E. Tanck. Towards clinical application of biomechanical tools for the prediction of fracture risk in metastatic bone disease. J. Biomech. 48:761–766, 2015.

    Article  PubMed  Google Scholar 

  17. Ding, M., A. Odgaard, and I. Hvid. Accuracy of cancellous bone volume fraction measured by micro-CT scanning. J. Biomech. 32:323–326, 1999.

    Article  PubMed  CAS  Google Scholar 

  18. Dragomir-Daescu, D., J. Op Den Buijs, S. McEligot, Y. Dai, R. Entwistle, C. Salas, L. J. Melton, III, K. Bennet, S. Khosla, and S. Amin. Robust QCT/FEA models of proximal femur stiffness and fracture load during a sideways fall on the hip. Ann. Biomed. Eng. 39:742–755, 2011.

    Article  PubMed  Google Scholar 

  19. Freeman, W. T., E. C. Pasztor, and O. T. Carmichael. Learning low-level vision. Int. J. Comput. Vis. 40:25–47, 2000.

    Article  Google Scholar 

  20. Gail, M. H. Systematic Error. In: Encyclopedia of Biostatistics. New York: Wiley, 2005.

  21. Genant, H., P. Delmas, P. Chen, Y. Jiang, E. Eriksen, G. Dalsky, R. Marcus, and J. San Martin. Severity of vertebral fracture reflects deterioration of bone microarchitecture. Osteoporos. Int. 18:69–76, 2007.

    Article  PubMed  CAS  Google Scholar 

  22. Gong, H., M. Zhang, Y. Fan, W. Kwok, and P. Leung. Relationships between femoral strength evaluated by nonlinear finite element analysis and BMD, material distribution and geometric morphology. Ann. Biomed. Eng. 40:1575–1585, 2012.

    Article  PubMed  Google Scholar 

  23. Goodall, C. Procrustes methods in the statistical analysis of shape. J. Royal Stat. Soc. B 53:285–339, 1991.

    Google Scholar 

  24. Grassi, L., N. Hraiech, E. Schileo, M. Ansaloni, M. Rochette, and M. Viceconti. Evaluation of the generality and accuracy of a new mesh morphing procedure for the human femur. Med. Eng. Phys. 33:112–120, 2011.

    Article  PubMed  Google Scholar 

  25. Hambli, R., and S. Allaoui. A robust 3D finite element simulation of human proximal femur progressive fracture under stance load with experimental validation. Ann. Biomed. Eng. 41:2515–2527, 2013.

    Article  PubMed  Google Scholar 

  26. Hambli, R., A. Bettamer, and S. Allaoui. Finite element prediction of proximal femur fracture pattern based on orthotropic behaviour law coupled to quasi-brittle damage. Med. Eng. Phys. 34:202–210, 2012.

    Article  PubMed  Google Scholar 

  27. Harrigan, T., and R. Mann. Characterization of microstructural anisotropy in orthotropic materials using a second rank tensor. J. Mater. Sci. 19:761–767, 1984.

    Article  CAS  Google Scholar 

  28. Hazrati-Marangalou, J., F. Eckstein, V. Kuhn, K. Ito, M. Cataldi, F. Taddei, and B. van Rietbergen. Locally measured microstructural parameters are better associated with vertebral strength than whole bone density. Osteoporos. Int. 25:1285–1296, 2014.

    Article  PubMed  CAS  Google Scholar 

  29. Hazrati-Marangalou, J., K. Ito, M. Cataldi, F. Taddei, and B. van Rietbergen. A novel approach to estimate trabecular bone anisotropy using a database approach. J. Biomech. 46:2356–2362, 2013.

    Article  PubMed  Google Scholar 

  30. Hazrati-Marangalou, J., K. Ito, F. Taddei, and B. van Rietbergen. Inter-individual variability of bone density and morphology distribution in the proximal femur and T12 vertebra. Bone 60:213–220, 2014.

    Article  PubMed  Google Scholar 

  31. Hazrati-Marangalou, J., B. V. Rietbergen, and K. Ito. Database of Femur Samples. Eindhoven: Eindhoven University of Technology, 2013.

    Google Scholar 

  32. Hulme, P., S. Boyd, and S. Ferguson. Regional variation in vertebral bone morphology and its contribution to vertebral fracture strength. Bone 41:946–957, 2007.

    Article  PubMed  CAS  Google Scholar 

  33. Johnell, O., and J. Kanis. An estimate of the worldwide prevalence and disability associated with osteoporotic fractures. Osteoporos. Int. 17:1726–1733, 2006.

    Article  PubMed  CAS  Google Scholar 

  34. Juszczyk, M. M., L. Cristofolini, and M. Viceconti. The human proximal femur behaves linearly elastic up to failure under physiological loading conditions. J. Biomech. 44:2259–2266, 2011.

    Article  PubMed  Google Scholar 

  35. Kersh, M. E., P. K. Zysset, D. H. Pahr, U. Wolfram, D. Larsson, and M. G. Pandy. Measurement of structural anisotropy in femoral trabecular bone using clinical-resolution CT images. J. Biomech. 46:2659–2666, 2013.

    Article  PubMed  Google Scholar 

  36. Kirkup, L., and R. B. Frenkel. Systematic Errors. In: An Introduction to Uncertainty in Measurement. Cambridge: Cambridge University Press, pp. 83–96, 2006.

  37. Kundu, A., S. K. Mitra, and P. Vaidyanathan. Application of two-dimensional generalized mean filtering for removal of impulse noises from images. IEEE Trans. Acoust. Speech Signal Process. 32:600–609, 1984.

    Article  Google Scholar 

  38. Larsson, D., B. Luisier, M. E. Kersh, E. Dall’Ara, P. K. Zysset, M. G. Pandy, and D. H. Pahr. Assessment of transverse isotropy in clinical-level CT Images of trabecular bone using the gradient structure tensor. Ann. Biomed. Eng. 42:950–959, 2014.

    Article  PubMed  Google Scholar 

  39. Lawrence, I., and K. Lin. A concordance correlation coefficient to evaluate reproducibility. Biometrics 45:255–268, 1989.

    Article  Google Scholar 

  40. Lekadir, K., J. Hazrati-Marangalou, C. Hoogendoorn, Z. Taylor, B. van Rietbergen, and A. F. Frangi. Statistical estimation of femur micro-architecture using optimal shape and density predictors. J. Biomech. 48:598–603, 2015.

    Article  PubMed  Google Scholar 

  41. Liu, X. S., X. H. Zhang, K. K. Sekhon, M. F. Adams, D. J. McMahon, J. P. Bilezikian, E. Shane, and X. E. Guo. High-resolution peripheral quantitative computed tomography can assess microstructural and mechanical properties of human distal tibial bone. J. Bone Miner. Res. 25:746–756, 2010.

    PubMed  PubMed Central  CAS  Google Scholar 

  42. Mc Donnell, P., P. Mc Hugh, and D. O’mahoney. Vertebral osteoporosis and trabecular bone quality. Ann. Biomed. Eng. 35:170–189, 2007.

    Article  CAS  Google Scholar 

  43. Modlesky, C. M., S. Majumdar, A. Narasimhan, and G. A. Dudley. Trabecular bone microarchitecture is deteriorated in men with spinal cord injury. J. Bone Miner. Res. 19:48–55, 2004.

    Article  PubMed  Google Scholar 

  44. Nishiyama, K. K., S. Gilchrist, P. Guy, P. Cripton, and S. K. Boyd. Proximal femur bone strength estimated by a computationally fast finite element analysis in a sideways fall configuration. J. Biomech. 46:1231–1236, 2013.

    Article  PubMed  Google Scholar 

  45. Oñate, E., J. Rojek, R. L. Taylor, and O. C. Zienkiewicz. Finite calculus formulation for incompressible solids using linear triangles and tetrahedra. Int. J. Numer. Meth. Eng. 59:1473–1500, 2004.

    Article  Google Scholar 

  46. Patel, T. K., M. D. Brodt, and M. J. Silva. Experimental and finite element analysis of strains induced by axial tibial compression in young-adult and old female C57Bl/6 mice. J. Biomech. 47:451–457, 2014.

    Article  PubMed  PubMed Central  Google Scholar 

  47. Pennec, X., P. Fillard, and N. Ayache. A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66:41–66, 2006.

    Article  Google Scholar 

  48. Pistoia, W., B. Van Rietbergen, E.-M. Lochmüller, C. Lill, F. Eckstein, and P. Rüegsegger. Estimation of distal radius failure load with micro-finite element analysis models based on three-dimensional peripheral quantitative computed tomography images. Bone 30:842–848, 2002.

    Article  PubMed  CAS  Google Scholar 

  49. Polgar, K., M. Viceconti, and J. Connor. A comparison between automatically generated linear and parabolic tetrahedra when used to mesh a human femur. Proc. Inst. Mech. Eng. H 215:85–94, 2001.

    Article  PubMed  CAS  Google Scholar 

  50. Polikeit, A., L. P. Nolte, and S. J. Ferguson. The effect of cement augmentation on the load transfer in an osteoporotic functional spinal unit: finite-element analysis. Spine 28:991–996, 2003.

    PubMed  Google Scholar 

  51. Rosipal, R., and L. J. Trejo. Kernel partial least squares regression in reproducing kernel Hilbert space. J. Mach. Learn. Res. 2:97–123, 2002.

    Google Scholar 

  52. Saha, P. K., and F. W. Wehrli. A robust method for measuring trabecular bone orientation anisotropy at in vivo resolution using tensor scale. Pattern Recog. 37:1935–1944, 2004.

    Article  Google Scholar 

  53. Schileo, E., F. Taddei, L. Cristofolini, and M. Viceconti. Subject-specific finite element models implementing a maximum principal strain criterion are able to estimate failure risk and fracture location on human femurs tested in vitro. J. Biomech. 41:356–367, 2008.

    Article  PubMed  Google Scholar 

  54. Seeman, E., and P. D. Delmas. Bone quality—the material and structural basis of bone strength and fragility. New Engl. J. Med. 354:2250–2261, 2006.

    Article  PubMed  CAS  Google Scholar 

  55. Sran, M. M., S. K. Boyd, D. M. Cooper, K. M. Khan, R. F. Zernicke, and T. R. Oxland. Regional trabecular morphology assessed by micro-CT is correlated with failure of aged thoracic vertebrae under a posteroanterior load and may determine the site of fracture. Bone 40:751–757, 2007.

    Article  PubMed  Google Scholar 

  56. Steiner, J. A., S. J. Ferguson, and G. H. van Lenthe. Computational analysis of primary implant stability in trabecular bone. J. Biomech. 48:807–815, 2015.

    Article  PubMed  Google Scholar 

  57. Tabor, Z., R. Petryniak, Z. Latała, and T. Konopka. The potential of multi-slice computed tomography based quantification of the structural anisotropy of vertebral trabecular bone. Med. Eng. Phys. 35:7–15, 2013.

    Article  PubMed  Google Scholar 

  58. Taylor, M., and P. J. Prendergast. Four decades of finite element analysis of orthopaedic devices: where are we now and what are the opportunities? J. Biomech. 48:767–778, 2015.

    Article  PubMed  Google Scholar 

  59. Tomaszewski, P. K., N. Verdonschot, S. K. Bulstra, and G. J. Verkerke. A comparative finite-element analysis of bone failure and load transfer of osseointegrated prostheses fixations. Ann. Biomed. Eng. 38:2418–2427, 2010.

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  60. Travert, C., N. Vilayphiou, H. Follet, and W. Skalli. Finite element vertebral model for fracture risk prediction: comparison of a full CT-based model versus two media simplified model, a preliminary study. Comput. Meth. Biomech. Biomed. Eng. 15:81–82, 2012.

    Article  Google Scholar 

  61. Varga, P., and P. Zysset. Sampling sphere orientation distribution: an efficient method to quantify trabecular bone fabric on grayscale images. Med. Image Anal. 13:530–541, 2009.

    Article  PubMed  CAS  Google Scholar 

  62. Wang, X., A. Sanyal, P. M. Cawthon, L. Palermo, M. Jekir, J. Christensen, K. E. Ensrud, S. R. Cummings, E. Orwoll, and D. M. Black. Prediction of new clinical vertebral fractures in elderly men using finite element analysis of CT scans. J. Bone Miner. Res. 27:808–816, 2012.

    Article  PubMed  PubMed Central  Google Scholar 

  63. Zou, W. W., and P. C. Yuen. Very low resolution face recognition problem. IEEE Trans. Image Proc. 21:327–340, 2012.

    Article  Google Scholar 

  64. Zysset, P., and A. Curnier. An alternative model for anisotropic elasticity based on fabric tensors. Mech. Mater. 21:243–250, 1995.

    Article  Google Scholar 

  65. Zysset, P., M. Ominsky, and S. Goldstein. A novel 3D microstructural model for trabecular bone: I. The relationship between fabric and elasticity. Comput. Meth. Biomech. Biomed. Eng. 1:321–331, 1998.

    Article  Google Scholar 

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Acknowledgment

The work of K. Lekadir was supported by a Juan de la Cierva research grant from the Spanish Ministry of Science and Innovation.

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Bert van Rietbergen is a consultant for Scanco Medical AG.

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Correspondence to Karim Lekadir.

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Associate Editor Karol Miller oversaw the review of this article.

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Lekadir, K., Noble, C., Hazrati-Marangalou, J. et al. Patient-Specific Biomechanical Modeling of Bone Strength Using Statistically-Derived Fabric Tensors. Ann Biomed Eng 44, 234–246 (2016). https://doi.org/10.1007/s10439-015-1432-2

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