title: Potential of in vivo MRI-based nonlinear finite-element analysis for the assessment of trabecular bone post-yield properties.
abstract: OBJECTIVE
  Bone strength is the key factor impacting fracture risk. Assessment of bone strength from high-resolution (HR) images have largely relied on linear micro-finite element analysis (μFEA) even though failure always occurs beyond the yield point, which is outside the linear regime. Nonlinear μFEA may therefore be more informative in predicting failure behavior. However, existing nonlinear models applied to trabecular bone (TB) have largely been confined to micro-computed tomography (μCT) and, more recently, HR peripheral quantitative computed tomography (HR-pQCT) images, and typically have ignored evaluation of the post-yield behavior. The primary purpose of this work was threefold: (1) to provide an improved algorithm and program to assess TB yield as well as post-yield properties; (2) to explore the potential benefits of nonlinear μFEA beyond its linear counterpart; and (3) to assess the feasibility and practicality of performing nonlinear analysis on desktop computers on the basis of micro-magnetic resonance (μMR) images obtained in vivo in patients.
  
  
  METHODS
  A method for nonlinear μFE modeling of TB yield as well as post-yield behavior has been designed where material nonlinearity is captured by adjusting the tissue modulus iteratively according to the tissue-level effective strain obtained from linear analysis using a computationally optimized algorithm. The software allows for images at in vivo μMRI resolution as input with retention of grayscale information. Associations between axial stiffness estimated from linear analysis and yield as well as post-yield parameters from nonlinear analysis were investigated from in vivo μMR images of the distal tibia (N = 20; ages: 58-84) and radius (N = 20; ages: 50-75).
  
  
  RESULTS
  All simulations were completed in 1 h or less for 61 strain levels using a desktop computer (dual quad-core Xeon 3.16 GHz CPUs equipped with 40 GB of RAM). Although yield stress and ultimate stress correlated strongly (R(2) > 0.95, p < 0.001) with axial stiffness, toughness correlated moderately at the distal tibia (R(2) = 0.81, p < 0.001) and only weakly at the distal radius (R(2) = 0.34, p = 0.007). Further, toughness was found to vary by up to 16% for bone of very similar axial stiffness (<2%).
  
  
  CONCLUSIONS
  The work demonstrates the practicality of nonlinear μFE simulations at in vivo μMRI resolution, as well as its potential for providing additional information beyond that obtainable from linear analysis. The data suggest that a direct assessment of toughness may provide information not captured by stiffness.
authors: Laboratory for Structural NMR Imaging, Department of Radiology, University of Pennsylvania Medical Center, 3400 Spruce Street, Philadelphia, Pennsylvania 19104, USA.
authors: Ning
authors: N
authors: Zhang
authors: Jeremy F
authors: JF
authors: Magland
authors: Chamith S
authors: CS
authors: Rajapakse
authors: Yusuf A
authors: YA
authors: Bhagat
authors: Felix W
authors: FW
authors: Wehrli
mesh_terms: D000368:Aged; D000369:Aged, 80 and over; D000465:Algorithms; D001696:Biomechanical Phenomena; D005260:Female; D020342:Finite Element Analysis; D006801:Humans; D007091:Image Processing, Computer-Assisted; D016014:Linear Models; D008279:Magnetic Resonance Imaging; D008875:Middle Aged; D013314:Stress, Mechanical; D013977:Tibia / Q000502:physiology*
references: J Magn Reson Imaging. 2007 Jul;26(1):191-7
references: 17659555
references: Magn Reson Med. 2009 May;61(5):1114-21
references: 19215044
references: Radiology. 2012 Mar;262(3):912-20
references: 22357891
references: J Orthop Res. 1999 May;17(3):346-53
references: 10376722
references: Bone. 2008 Jun;42(6):1203-13
references: 18358799
references: J Biomech. 2000 Dec;33(12):1575-83
references: 11006381
references: Magn Reson Med. 1999 May;41(5):947-53
references: 10332878
references: J Bone Miner Res. 2008 Mar;23(3):392-9
references: 17997712
references: Ann Biomed Eng. 2011 Feb;39(2):742-55
references: 21052839
references: PLoS One. 2012;7(4):e35525
references: 22558164
references: Osteoporos Int. 2005 Mar;16 Suppl 2:S25-35
references: 15340800
references: J Orthop Res. 1998 Sep;16(5):622-8
references: 9820288
references: J Biomech. 1994 Sep;27(9):1137-46
references: 7929463
references: Top Magn Reson Imaging. 2002 Oct;13(5):323-34
references: 12464745
references: J Bone Miner Res. 2002 Sep;17(9):1629-38
references: 12211433
references: Drug Discov Today Technol. 2006 Summer;3(2):221-9
references: 24980411
references: Bone. 2009 Apr;44(4):579-84
references: 19135184
references: Ann Biomed Eng. 2012 Jul;40(7):1575-85
references: 22258889
references: Bone. 2010 Sep;47(3):556-63
references: 20685323
references: J Clin Invest. 1988 Jun;81(6):1804-9
references: 3384952
references: J Biomech. 2004 Nov;37(11):1671-8
references: 15388309
references: Magn Reson Med. 1996 Jun;35(6):903-10
references: 8744019
references: Bone. 2011 Oct;49(4):895-903
references: 21784189
references: J Biomech. 1999 Jul;32(7):673-80
references: 10400354
references: Bone. 2006 Dec;39(6):1218-25
references: 16904959
references: J Biomech. 2001 May;34(5):569-77
references: 11311697
references: Med Eng Phys. 2007 Dec;29(10):1096-105
references: 17229586
references: J Biomech. 1995 Jan;28(1):69-81
references: 7852443
references: J Biomech. 2004 Jan;37(1):27-35
references: 14672565
references: Bone. 1994 Jan-Feb;15(1):105-9
references: 8024844
references: Comput Methods Biomech Biomed Engin. 2008 Aug;11(4):389-95
references: 18568833
references: Osteoporos Int. 1995 May;5(3):167-73
references: 7655177
references: J Biomech. 2008;41(7):1479-85
references: 18423473
references: Philos Trans A Math Phys Eng Sci. 2010 Jun 13;368(1920):2653-68
references: 20439267
references: Bone. 1996 Apr;18(4):327-30
references: 8726389
references: J Bone Miner Res. 2002 Apr;17(4):687-94
references: 11918226
references: Bone. 2002 Feb;30(2):404-11
references: 11856649
references: Bone. 2011 Jun 1;48(6):1232-8
references: 21376150
references: J Magn Reson Imaging. 2010 May;31(5):1157-68
references: 20432352
references: Bone. 2003 Oct;33(4):457-65
references: 14555248
references: Med Eng Phys. 2001 Apr;23(3):165-73
references: 11410381
references: IEEE Trans Med Imaging. 2005 Dec;24(12):1574-85
references: 16353372
references: JAMA. 1990 Feb 2;263(5):665-8
references: 2404146
references: J Bone Miner Res. 1994 Aug;9(8):1137-41
references: 7976495
references: Biomed Mater Eng. 2004;14(3):303-10
references: 15299242
references: J Biomech. 1998 Jul;31(7):601-8
references: 9796682
references: Acta Biomater. 2006 Jan;2(1):59-68
references: 16701859
references: J Bone Miner Res. 2008 Sep;23(9):1426-34
references: 18410234
references: Bone. 2002 Jun;30(6):842-8
references: 12052451
references: Bone. 2003 Oct;33(4):494-504
references: 14555252
references: Stud Health Technol Inform. 1997;40:97-112
references: 10168885
references: Calcif Tissue Int. 1993;53 Suppl 1:S7-13
references: 8275383
references: J Bone Miner Res. 2010 Jun;25(6):1406-14
references: 20200948
references: NMR Biomed. 2006 Nov;19(7):731-64
references: 17075953